What is a fugue?

Answer 1

Think of fugues like rounds (row, row, row your boat) but instead of the subsequent voice coming in on the original pitch, it comes in on a different pitch (sometimes a third above or below the original). Four voice texture is common, and, unlike a round, fugues don't usually last like that through the whole piece --it's more of a musical device than something that happens the whole way.
There are LOTS of Bach fugues. Listen to a couple online and you'll get it.

Answer 2

Fugue is either a composition or a musical treatment of material using a subject (short main theme) followed by an answer in another voice (usually in a different octave or in a related key). While the other voices take up the subject, a countersubject is heard followed by a series of episodes which expose different properties or fragments of the subject.

Usually, fugues have three or four voices and more or less maintain this texture throughout the piece.

Fugue is most associated with Baroque music, especially Bach but composers from nearly all periods have tried their hand at fugue writing.

Answer 3

A fugue is a composition where a theme, or "motif" is stated at the beginning and repeatedly interwoven throughout the song in different voices. An example of this might already be on your computer: the second (scherzo) movement of Beethoven's Ninth Symphony is a fugue.