Concerning Christianity, all Jesus really implies in the canonical scriptures is that the Kingdom is coming, and we should all prepare ourselves; he doesn’t come out and explicitly say anything about what we should expect. Not a whole lot there that’s new. Jesus seems to expect the usual heaven/hell dichotomy—the virtuous going to the former, the sinners cast into the latter. "Hell," here, is gehenna in Latin, geenna in Greek transliteration from Hebrew, and it’s just “the valley of Hinnom outside Jerusalem (often, but erroneously, translated in English as hell)…the place where the Canaanites were said to have been sacrificing their children to their god Moloch (Molech in Hebrew).” By the time Jesus was around, it was a dump for the corpses of criminals and also for, well, garbage! Fires were set to burn all the lovely refuse, and sometimes sulfur was added to sanitize the place and keep the fire burning. Read more about this in Mark, which goes along with Isaiah about the whole geenna prophecy. Jesus himself didn’t really stress the punishment bit so much as the salvation-eternal life gig.

Jesus equates geenna with not only that fun image, but also with the total destruction of a person, body and soul—the real kicker here being the oblivion of your mortal soul, which makes entering the Kingdom of God impossible. However, the guy who ignored the beggar Lazarus found himself in Hades, "burning from thirst and begging for a drop of water in great agony, while the poor man was seen in the embrace of Abraham. A great gulf separated the places were either of them was seated, to prevent crossings from one side to the other (Luke 16:19-31).”

That image seems to contradict the first bit, regarding oblivion, but it also goes along with Greek conceptions of refreshments being offered to the virtuous dead; Tantalus comes to mind as the Greek concept of what happens to the impious dead as far as food goes, standing in water he can’t drink and looking at fruit he can’t eat.

One last bit of contradiction in eschatology here: When does this judgment happen? It’s either immediately after the person’s death or during the second Coming, but it’s never made explicitly clear:

Although most of the dead would have to await their resurrection to be judged according to their deeds (Matt 25:31-46), some few could expect a special treatment. Most conspicuous is the case of the repentant criminal on the cross who was granted immediate entry to paradise (Luke 23:40-43), an idea that makes sense under the circum stances and is also in line with Jesus' attitude towards outcasts who acknowledge their sins in earnest. Some early Christians understood, with good reason, that this promise was also applicable to martyrs. (Kyrtatas)

A lot of the ideas of Hell seem to be derivative from Hebrew scripture and Hellenistic myth—oh! Akhmim fragment! Apocalypse of Peter!

It’s fun, it’s apocalyptic (all the best imagery is), it’s got Hellenistic overtones. The basic idea is that, rather than the canonical Revelation version of the Last Judgment (read: that the righteous live on and the sinners meet a final death), a Dante-esque categorization of the dead sinners will take place, and they’ll be meted out punishment specific to their crimes.

The Akhmim fragment is far removed from its original composition (read: about four centuries), and it’s in Greek. Kyrtatas’s paper (listed in the sources) has a lot of interesting history about the story’s constant reworking to incorporate new ideas, new punishments, et cetera, but here’s the bit from Bremmer’s paper on Paul and Peter:

The Apocalypse of Peter is highly important in the history of hell, as it was "the first major Christian account of postmortem punishment out side the New Testament." Unfortunately, we cannot be totally certain about its date and place of origin. In the end we are left with two conclusions. First, it was first written in Palestine but revised by a Jewish Christian author who used an Egyptian source or version and clearly was well-educated and at home in Greek culture. This orientation is also supported by the fact that he used the Septuagint for his quotations of the Old Testament. Alternatively, we also cannot exclude that it was written by a similarly educated author in Egypt using a Palestinian text. At this point, we simply do not know (Van Ruiten 2003). On the other hand, its date is much less disputed. At present there is a general consensus that it must date from the last decades of the first half of the second century ad, given its mention by Clement of Alexandria (apud Eus. HE 6.14.1; cf. Lightfoot 2007:132).

Bremmer goes on to mention Aristophanes, Plato (or pseudo-Plato), Vergil, and Greek myths like Sisyphus (best known for his hit, “rolling boulder up a hill for eternity") and Prometheus (having his internal organs pecked out, regrown, pecked out again. For eternity).

Here are the greatest hits of the Akhmim fragment hell, translated by yours truly I CRINGE, but it's already on my laptop:

[VII-VIII] Some sinners were hanging by their tongues--blasphemers of the path of the just, lying upon a bed of fire as punishment. A great lake full of burning filth pooled there, in which stood those who turned from righteousness; angels fell upon them as torturers.

[IX]And there were others; women, adulteresses, were hung by the locks of their hair, above the boiling filth. And the men who commingled with them in the stain of adultery were hanging by their feet, and their heads were buried in the mire, and they cried with great shouts, "We did not believe we would be sent to this place!"

[X] And I saw murderers (and their conspirators) being thrown into a place full of horrors and wicked creeping things, being gored by wild beasts and writhing in torment. And upon them were worms, just as clouds of darkness. And the souls of those who were murdered were standing above, overseeing the punishment of each of the murderers, and they proclaimed, "God, your judgment is lawful."

[XI] And near this place, I saw another, where the pus and the foulness of those being punished was running down, forming something like a lake. There sat women, up to their necks in this mixture; opposite them were many children, brought into the world in an untimely manner, seated and crying. Emitting from the infants were gouts of fire, and the flames were striking the women about the eyes. These women were the unmarried who brought fetuses into the world and wounded them.

It basically goes on to list a number of punishments for a corresponding number of sins (including usurers, false martyrs, and homosexuals), and I’ve included a link in the sources for further reading!

Edit: Grammar.

Sources/Further Reading:

Here, Fiensy’s paper on lex talionis in the “Apocalypse of Peter”

Here, Bremmer's paper on Christian apocalypse..s?

Here, Kyrtatas’s “Origins of Christian Hell”

Here, for a public domain full translation of the Akhmim fragment, but it’s not the best translation I’ve ever seen