Great question, that's actually my field of work :)

For beginners, one thing to note in such reconstructions is the use of brackets. Text between ( and ) is emended from an abbreviation that is present in the actual text. As a modern example: N(orth) A(tlantic) T(reaty) O(rganization). Square brackets '[ ]' indicate text that has been lost, and possibly reconstructed by the editor. [[ ]] is for erased text, [[[ ]]] for text inscribed into an erasue. {} is for redundant text (doublettes for example). In [...] the dots signify that that passage is lost, but the editor is sure that three letters were there, [---] is for an uncertain number of lost letters, and [------] for lost lines. '?' is for readings where the editor is uncertain. There are more, special and specific signs, like < and> signifying original errors corrected by the edi<L=t>or, + for unidentifiable single letters, and so on - this is called the Leiden bracket system.

There are several methods employed to emend the lacunae and abbreviations. In the above example, the first line is easily emended, because the dative ending is the only possible ending since Mars is already in the dative case, as fits a dedication. The following is also pretty easy, since you can at first glance see that it is the name of an emperor - the IM --- AR DIVI NERVA[E] F... part gives that away with 100% certainty. So you can easily emend to Im[perator Caes]ar...; we also now know that the whole title stood in the nominative case. The rest is pretty easy, since the emperors names were very formulaic and almost always written the same way. Now, to pinpoint the exact title used here, we need a bit more data. The titles changed over time, new titles were added and the offices they held were counted. So as a next indication, we have "]T XIII" in line 6, where we could reasonably expect a TRIB POT followed by a number, counting the times the emperor had been awarded the tribunicial powers, which occured every year as a matter of fact (the tribunicia potestas annua), usually on 10. December. So we have a number for that, which is great, because we could then, in the absence of other evidence, precisely (relatively speaking, but a possible space of two years is mathematical precision for our profession) date this inscription! We can, with confidence, then emend that line to "Trib(unicia) pot(estate) XII". Another indication for dating would be the "V P P" in the next line. Before the p(ater) p(atriae), we would expect his consulships. So Co(n)s(ul) V p(ater) p(atriae)comes into this line.

With this information, we can consult other evidence, and see that Trajan was awarded his tribunician powers for the 12th time in December 108, which lasted until 9.12.109. He was consul for the fifth time in January 103, and for the 6th time in January 112. So we can date this to some time between 10.12.108 and 9.12.109. This makes it easy to fill in the other titles. He would have been awarded the title imperator six times by then, and have aquired the triumphal names germanicus, victor over the Germans (Nov. 92) and dacicus (Autumn 102), but not yet parthicus (February 117). From here on, it's trivial to reconstruct the full title, because we can fill in the appropriate titles and numbers for that time (if they fit the available space). If we couldn't precisely date them, we would just add the titles he had for sure without a number.

The next part is where it gets interesting, because we only have ITU, D and RUM (and E in the last line). If we consult the available evidence, we see that the monument here was a Tropaeum, erected as a triumphal monument, probably for a successful campaign. If we look at the historical context, we see that Trajan won his war against the Dacians under King Decebal just the year before, and Decebal had died - an event also celebrated on the column of Trajan. So we can reasonably expect the monument mentioning the reason why this was erected in honor of the God of War by the emperor. And this could have been "victo exercitu dacorum ... et sarmatarum", after he defeated the army of Dacians and Sarmatians. Another possibility fitting the available evidence would by "victo exercitu d[ecebali daco]rum", after he defeated the army of Decebal of the Dacians. So there's always an element of uncertainty, which the editor in the above example signified with the '?', because he probably knew that there were other possibilities.

So this was one example. The great thing about Roman inscriptions is that they are usually very formulaic, and you usually know what element you would expect in the lacuna, even if it is quite large. What's true for an emperor is also true of his subjects - the name followed usual formulas. This isn't different from today, when you know someone named Prof. Dr. Steven R. Ingleschneck III, and found a letter on your desk that was damaged in some way and you could only read "....r.....ve......ng..lesch......" you'd probably immediately recognize what the full name would be. In the same way, Roman names usually followed a formula. Full citizens had the tria nomina, the full name containing the praenomen, first name, nomen gentile, family name and cognomen, a nickname, together with the name of the father, the voting tribe and the origin. So a Roman citizen's full name could for example be Marcus Aemilius Marci filius Stellatina tribu Rufus Torino - Marcus Aemilius, son of Marcus, called "the Red" of the voting tribe 'Stellatina' from Turin. Some also added grandfather and great-grandfather to signify their long tradition and nobility. Soldiers usually continued with their rank and the units they served in, nobles or officials and officers would add their cursus honorum, the honors and titles they won and held during their career. Freedman would add their liberator, slaves their master, priests which collegium they belonged to, artisans their craft, and so on and so forth. I should probably add this is only the ideal type. Since Roman Citizenship lost its value over time, people didn't need it to signify status anymore and didn't use the full name; provincials or foreigners might follow different naming practices; customs changed over time.

But not only names, whole inscriptions followed certain usual formulars. A stereotypical votive inscription, honoring a deity and thanking them for services rendered, would begin with the name of the god or gods in the dative case, followed by the name(s) of the dedicants in the nominative with appropriate titles, possibly the reason for this dedication (for example, 'ex visu', because of a vision) and a closing formula that the dedication was given freely and willingly because they were satisfied. A funerary inscription began with the name of the deceased and his appropriate titles, ranks and units, his age at death (and length of military service), sometimes with the cause of death given, possibly but rarely some more elaborate text or poem, possibly the names of other persons interred here, followed by the name of the one who erected the monument (sometimes the cause, for example 'because of the testament', is given, and often the relation to the deceased or terms of endearment like 'beloved wife' 'best and rarest husband'). Building inscriptions usually contained a dedication to the emperor, what was repaired or built, in military constructions the unit who performed the construction, and the one responsible for execution and approving the building.

Now this doesn't mean that we can always reconstruct everything fully - far from it. But we can often get a general gist of what type the monument was, and from that reconstruct with approximate precision from the surrounding evidence, such as ornaments or reliefs, historical, archaeological and spatial context what might probably have stood there, or at least what type of things would have stood there, even from very minor fragments.

This is far from a perfect science. The praxis of enclosing reconstructions between square brackets has given rise to the term "history from square brackets", originally coined by E. Badian. Many scholars fall into the trap of reading the edited material and take the text between [ and ] for as accurate as the rest of the source, and if the reconstrucion was uncertain and there is no other corroborating evidence around, you're building your thesis, often quite literally, on sand. Which, of course, doesn't deter everyone because it is a) tempting, b) easy and c) often the only evidence at all for something (the Augsburg Victory altar, for example, because of a few little lines of text, made us get a clear picture of a significant period of years that was completely in the dark for us before.) This means that autopsy is often the only reliable way to work with inscriptions, since when you see it with your own eyes you become acutely aware of how uncertain it all can be, though modern digital media thankfully will change this in the future. I am so incredibly excited for 3D-printing inscriptions I should probably sacrifice at least an oxen to Minerva...

In the end, it is an imperfect method. We can often be 100% sure, and will as often be at a complete loss, but most guesses are very educated and new ones all get peer-reviewed pretty quickly. Even so, there are many old readings around who haven't yet been subject to the eye of the modern discipline.

There are other important things to consider, like if the reconstructed lines fit in the available space, which often can be reconstructed or guessed at as well, things like the layout of lines and inscription fields, and metric for verse inscriptions that can help in a reconstruction, but this is the general idea. Experience helps a ton, too. If you have more questions, I'd be glad to answer :)

Edit: corrected some spelling errors

Others have covered Roman epigraphy and an analogy for what it would be like in English. Maybe I can tackle a small slice of Greek papyrology, as it stands a slight but real chance of getting into the news sometime in the next few years: just in January this year, it was announced that some progress has been made with a non-intrusive technique for mapping the position of ink in carbonised Herculaneum papyri, from the philosophy collection of L. Calpurnius Piso Caesoninus which got crushed and burned in the eruption of Vesuvius in 79 CE. It's got a long way to go yet -- so far they've managed to read a few letters from near the middle of what is in effect a solid lump of coal, using X-ray phase-contrast tomography -- so papyrologists are still going to have plenty to occupy them with trying to fill in the gaps, even once the method is greatly improved.

But 1st century philosophy isn't my area: it's early Greek poetry, especially epic. I've been gathering an archive of early epic poetry for personal research purposes over the last week, and about a third of the extant corpus (not including Homer) comes from fragmentary papyri. As a sample, here's a passage from a papyrus of the Hesiodic Catalogue of Women, as it stands in the surviving bits of the papyrus:

....]ουρεια[.......]θεαιεξεγενοντο
....]ενοσου[.........]τυρωνκαιαμηχανοεργων
....]ητεστε[..........]ιγμονεσορχησ[
ξουθοσδεκ[...........]ρατονειδοσεχ[
κουρηνκαλλ[..........]χθηοσθειοιο
.....]τωνι[.............]ησατακ[..]τιν
....]χαιονεγ[...............]τοπωλ[.]ν
....]εισεν[..............]ιδεαδιομηδην

So, how intractable a situation is this? Well, it depends on a lot of factors. In this passage it's possible to fill in gaps based on three types of consideration.

First consideration. The papyrus has no punctuation, no spaces between words, and no accents, so those have to be supplied by someone who is familiar with the language and can recognise words. To some extent this can also allow filling in words beyond the frayed papyrus-edge if they're obvious enough: this gives

....] οὔρεια[ι ......] θεαὶ ἐξεγένοντο
....]ενοσου[...... Σα]τύρων καὶ ἀμηχανοέργων
....-]ητές τε [..........-]ιγμονες ὀρχησ[τῆρες
Ξοῦθος δὲ κ[-...........-]ρατον εἶδος ἔχ[-
κούρην καλλ[ι-..........-]χθηος θείοιο
.....-]των ι[-.............-]ησατ' ἄκ[οι]τιν
... Ἀ]χαιὸν ἐγ[-...............]τοπωλ[.]ν
....]εισεν[..............-]ιδεα Διομήδην

And now there are a few interpretible phrases: e.g. line 1 "there were born mountain-dwelling... goddesses", line 2 "and of wastrel Satyrs", and so on.

Second consideration. Supplementary testimony can fill gaps. A surviving ancient source happens to quote part of line 1 and all of lines 2-3, so that bit can be filled in trivially. I won't quote the whole passage again just yet.

Third consideration. Once we know what the passage is about, we can fill in words based on context. If the passage is talking about Satyrs and goddesses, and mentions "Xouthos" and "Achaios" (a legendary son and grandson of Hellen), we can compare information we already have from other sources -- in this case, about the mythological content -- and use that to fill in gaps. This gives:

ἐξ ὧν οὔρειαι Νύμφαι θεαὶ ἐξεγένοντο
καὶ γένος οὐτιδανῶν Σατύρων καὶ ἀμηχανοέργων
Κουρῆτές τε θεοὶ φιλοπαίγμονες ὀρχηστῆρες.
Ξοῦθος δὲ Κ[ρείουσαν ...]ρατον εἶδος ἔχ[ουσαν
κούρην καλλ[ι-...... Ἐρε]χθῆος θείοιο
.....-]των ι[-.............-]ησατ' ἄκ[οι]τιν
... Ἀ]χαιὸν ἐγ[...............]τοπωλ[.]ν
[μιχθ]εῖσ' ἐν [..............-]ιδεα Διομήδην

In line 4, Kreiousa is Xouthos' wife, and we know her father was Erechtheus. That's got the additional benefit of being 100% compatible with line 6 ἄκ[οι]τιν "wife". Achaios and Diomede are their children. Ion also ought to be listed as their brother, so it'd be nice to fit him into one of the gaps; thatt can't be in the last line, which talks about Xouthos and Kreiousa having sex (μιχθ]εῖσ') and then Diomede, with something that looks like an adjective referring to Diomede. So Ion must be in the gap in the middle of the second-last line.

Fourth consideration. To fill in the rest, you need something that won't be applicable to other fragmentary texts: because it only works in the context of archaic epic poetry, it's in a way especially interesting. And this is: a detailed knowledge of the system of metrical formulae that poets used as the building blocks of their poetry. If you have a concordance, or a file-index of metrical formulae, or -- and this is what I've been putting together over the last week -- an easily searchable digital compilation of all early hexameter verse, you'll be able to plug in distinctive words and see which metrical formulae those words regularly appear in.

For example, take the phrase in line 4 εἶδος ἔχ[ουσαν, "possessing beauty". If we search Archaic hexameter poetry for this phrase, we'll find the following lines that use the same formula:

Theog908: Ὠκεανοῦ κούρη πολυήρατον εἶδος ἔχουσα
Hy2Deme315: Δήμητρ' ἠύκομον, πολυήρατον εἶδος ἔχουσαν
Hy12Hra2: ἀθανάτην βασίλειαν, ὑπείροχον εἶδος ἔχουσαν
Hy32Sel16: ἐκπρεπὲς εἶδος ἔχουσαν ἐν ἀθανάτοισι θεοῖσι
CatF17a_7: [.............]..ν πολυήρατον εἶδος ἔχουσ[αν]
CatF25_39: γείνατο δ' Ἰφιάνειραν ἐπήρατον εἶδος ἔχουσα[ν]

(I omit a few results that would be cheating for the purpose of this exercise.)

With these results, we can see if we can reasonably fill in a longer form of the formula. For that purpose we focus on the ones in the same metrical position -- at the end of the line -- and find that there are three adjectives agreeing with εἶδος that could be filled in: πολυήρατον (3×), ὑπείροχον (1×), and ἐπήρατον (1×). In fact the second and third of these are allomorphs of the first one, since they represent adjustments of the same formula for a different metrical context: πολυήρατον is used in lines which have a type of prosodic structure called a "hepthemimeral caesura"; the other two are used in lines with a "tritotrochaic caesura".

With Κ[ρείουσαν filled in in the first half of the line, we're talking tritotrochaic caesura, so it's going to be ὑπείροχον or ἐπήρατον. Only ἐπήρατον fits. Even if ὑπείροχον were possible, the formulaic system has such a strong effect in this corpus of poetry that, looking at other details (which I'll omit here), there's no real doubt about it.

We can then go on to use the same procedure for the other gaps, and in this case it turns out that the entire passage can be confidently filled in based on metrical formulas:

ἐξ ὧν οὔρειαι Νύμφαι θεαὶ ἐξεγένοντο
καὶ γένος οὐτιδανῶν Σατύρων καὶ ἀμηχανοέργων
Κουρῆτές τε θεοὶ φιλοπαίγμονες ὀρχηστῆρες.
Ξοῦθος δὲ Κ[ρείουσαν ἐπή]ρατον εἶδος ἔχ[ουσαν
κούρην καλλ[ιπάρηον Ἐρε]χθῆος θείοιο
ἀθανά]των ἰ[ότητι φίλην ποι]ήσατ' ἄκ[οι]τιν,
ἥ οἱ Ἀ]χαιὸν ἐγ[είνατ' Ἰάονά τε κλυ]τόπωλ[ο]ν
μιχθ]εῖσ' ἐν [φιλότητι καὶ εὐε]ιδέα Διομήδην.

From them were born mountain Nymphs, goddesses,
and the race of Satyrs, good-for-nothings and wastrels,
and the gods (called) Kouretes, who love fun and dancing.
And Xouthos married Kreiousa, who possessed lovely beauty,
the fair-cheeked daughter of divine Erechtheus,
dear, by the will of the gods; he made her his wife.
She bore to him Achaios and Ion, famed for horses,
having sex with (Xouthos); and also beautiful Diomede.

Filling in the lacunae is indeed partly guesswork. But it is very educated guesswork. We have many other examples of texts, we know the language. It's kind of like filling in the blanks on a form letter. We have similar inscriptions, which allows us to know the kind of inscription it is, and gives us a hint as to what it might contain. Then, we know the language, so it's about filling in the gaps with what makes sense.

For example, let us take something most Americans are familiar with, the Constitution.

We th ople o he Un ates, in O r to fo a more perf Union, est h J tice, ins estic Tranqu provide f ommon defen omote the gene lfare, an ecure the Bl gs of Lib y to selves a r Poste o or n an ablish th titution f e Un ates of Ame

If you know the language, and you recognize some of the forms, you can fill in the blanks intelligently. That's part of it. Most of the work relies on those two things. An inscription like this is a sort of formal description of titles, which follows a recognized form. We know the names and the titles, and so we can recognize them in their parts and fill in the blanks.

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Edit: formatting and looks like the spacing is weird when it publishes the post. Oh well.

Many of these word/acxronyms or whatever are used a lot on many, many different objects/sources from that time, and an exemple of this is on coins. You'd have coins with many of these shortened words on them, and in the end, I don't know the exact methodology used in order to define exactly what every single one of these words parts means, but they appear so often that professionals who work on that kind of thing know it quite easily... I'm not an expert personnally of roman, greek or other civilizations from the Antiquity, but I've seen many of my university teachers easily decyphering these acronyms or shortened words.

It definitely isn't guess work, they know exactly what they're doing.

I'll add a few notes for cuneiform texts, which have been alluded to. Many of the methods described already apply here as well, but some additional considerations present themselves. First, when looking at a fragmented tablet we can find if it has a shape associated with a certain date and type of text. Some shapes are very strictly associated with certain text types. While translating scholars can then attempt to look for joins based either on the shape of the tablet(is there another bit of this someplace else?) or other known fragments that seem to pick up where the text at hand leaves off. In some cases one might find that a published fragment supplements the one under study. In addition, one can determine if the text is part of a known work and if so if it can be supplemented with other manuscripts of the same text. Finally, many of the techniques of finding known patterns apply here; for instance many royal inscriptions are very formulaic and one can predict the patterns they will follow.