In Latin, “bachelor” is baccalaureus (or baccalarius). Flattering themselves, medieval scholars thought it came from the phrase bacca lauri, which means “laurel berry,” since the bachelor’s degree was a mark of honor, just like the laurel wreath that crowned ancient athletes and poets, as in “poet laureate.” But the true etymology of the word is pretty much unknown; it may have something to do with cows (vacca in Latin, where that "v" can sound like "b" in some Romance languages) but that's very suspect. More probably it's related to a land measure in the early Middle Ages called a baccalaria, so the peasants who assisted in working it were baccalarius/ia. In later medieval use, “bachelor” meant, first, a young knight without land, then later a junior member of a guild. Its sense of “unmarried person” came from the notion that a bachelor was young and inexperienced, just starting out and not yet established, and therefore not apt to be married. In Latin, universitas and collegium simply meant any organized group; a guild of wool merchants or a guild of bakers could be called a collegium or universitas, which is why modern colleges and universities are names such; in fact, we still use the word for something like the US electoral college. So the medieval university was essentially a guild of master scholars and their apprentices. In the medieval university, those who gained the bachelor’s degree--a term first appearing in Paris in the 13th-C--were well enough educated to be apprentice teachers but not yet masters. But the origin of the word and its evolution are pretty hazy.
EDIT: This explanation is from my old lecture notes, but you can get a more etymologically involved summary of the word from the Oxford English Dictionary entry for "bachelor," which most schools should have available online.

EDIT 2: I decided to add the bit about the meaning of collegium/universitas to make clearer the link between other apprentice bachelors.