(Note that Bach's "Toccata and Fugue" is two separate sections; if you're looking for fugal writing in the toccata portion, you'll have some trouble!)
Fugue is a type of polyphonic contrapuntal writing (the noun is "counterpoint," and "polyphonic" means "many voices"). "Counterpoint" basically just means the relationship among written notes; it stems from the Latin punctum contra punctum, or "note against note." (The connection is more obvious in the German Kontrapunkt.)
Fugues begin with an exposition where the subject (and possibly countersubject) are introduced in each voice. Once all voices have entered with the subject, the fugue enters into one or more episodes where they develop thematic material, move through extra keys, and so on.
Types of development include elongating the (counter)subject, shortening it, flipping it upside down, playing it backwards, using only a section of it, playing it in multiple voices at the same time (but slightly staggered), some combination of all of these, etc.
There's a famous statement that "fugue is not a form, it's a process." This is because the possibilities are really limitless in terms of keys visited, types of development employed, etc. Fugues can be remarkably different, which is why it can be so difficult to explain a fugue in one sentence.
Dom's answer here is really terrific and should definitely be consulted.
And, if you're interested, check out what I think is the single most impressive example of fugal writing of all time.
Lastly, two tongue-in-cheek references: