You may want to ask at /r/linguistics. Studies of the history of language largely come from people trained as linguists. As mentioned by /u/comrade_chernov, most languages in Europe probably have a common ancestor. There are enough written historical records to determine that the Romance languages developed from Latin and to estimate how fast changes generally take place in language. In most other cases, the proto-language has to be reconstructed using clever guess-work. The name given to the supposed ancestor of the whole family is "Proto-Indo-European" (PIE). Several guesses have been made about the peoples who spoke this language. It seems clear that it originated in the East, possibly in what is now Turkey or the former Soviet Union.

Aside from Germanic, Romance, Celtic and Slavic languages, there is also IndoIranian. This group includes the Kurds, Iranian (Persian), Hindi in India and a number of other major languages.

In some cases, similarity is based on geographical and cultural proximity and not ancestry. In other words, words and even sometimes sounds and possibly grammatical features were borrowed from other languages with which people came into contact. The words "alcohol" and "algebra", which are similar in many European languages, were borrowed from Arabic, which is not a descendant of PIE. Two of the synonyms for "book" in Indonesian, "buk" and "kitab", were borrowed from Dutch and Arabic respectively, although its ancestor is neither PIE nor the ancestor of Arabic.

This borrowing often happens when a language has no words of its own for a concept. Arabic science and mathematics were more advanced than that of Europe when "algebra" and "alcohol" were borrowed. Similarly, many sub-branches of Indo-European borrowed a lot from Latin because of the power of the Roman empire.

The words that are seldom borrowed and can be reliably be used to determine whether languages share an ancestor are those that are used for very simple, low-technology concepts, like family relationships, words describing nature like"tree" or "water", and simple numbers. You should be able to find some comparison tables for Indo-European vocabulary with google. Often either the meaning or the sound or both have changed, so don't expect everything to be obviously similar (e.g. a common find is that in the Germanic branch, the word begins with H, in Romance, with C and in Slavic with S)

Examples (just from those languages I can remember)

Germanic: hundred, honderd, hundert

Romance: centum, cent

Slavic: sto

Sanskrit: satem

---

Germanic: sun, son, Sonne

Romance: sol, soleil

Slavic: solntse

---

Germanic: fish, vis, fisk, Fisch

Romance: pisces, poisson

Slavic: ryba

---

Germanic: brother, broeder, broer, Bruder

Romance: frater, frère

Slavic: brat'

I'm by no means a professional linguist, but I'll give as best a rundown as I can.

In Europe there are three primary language families - the Romance, Germanic, and Slavic families. There are of course exceptions and lesser-spoken languages, but these three are the ones which make up the vast majority of European society.

Romance languages are those coming from former portions of the Roman Empire which developed out of vulgarization/local "slang" dialects of Latin; Italian, French, Spanish, Portuguese, Romanian, and various smaller regional languages and dialects such as Romansh and Galician, as well as modern Latin itself.

Germanic languages, as the name suggests, stem from the ancient Germanic tribes, who shared a common language known as proto-Germanic. Over time, as these groups spread out, their dialects became separate languages. These include German, Dutch, English, Swedish, Norwegian, Danish, Icelandic, as well as many regional dialects such as Scots, Frisian, and Faroese.

Slavic languages are the "newer" of the three and come from the Slavic peoples from late antiquity/early medieval times. I describe them as "newer" because if I remember correctly proto-Slavic/Slavonic was a single language until nearly 1000 AD, at which point it diverged into languages such as Russian, Ukrainian, Belorussian, Bulgarian, Polish, Czech, Slovene, Slovak, Serbo-Croatian...many of the "Eastern European" languages, plus many regional dialects. From what I understand there is a much higher degree of mutual intelligibility within Slavic languages than within Romance or Germanic, owing to the more recent time that it split apart from proto-Slavic/Slavonic.

Some outliers here are the Celtic language family (Irish and Scottish Gaelic, Welsh, Manx, Cornish, and Breton), Greek, which is a part of its own language family (Hellenic), Basque, which if I recall correctly people aren't sure of the origins of, and Hungarian and Finnish, which are actually both Finno-Ugric languages; the Hungarians (Magyars) were a tribe which came down from that area and settled in central Europe starting in the 8th century.