What exactly is bourbon?

In short and succinctly:
Bourbon is a whiskey that comes from the USA, has at least 51% corn and is stored in new oak barrels for a minimum of 3 years.

The new barrels give it the characteristic vanilla notes. These arise from the roasting of the barrels. (All barrels are burnt out to a greater or lesser extent on the inside, or "toasted") In combination with the high proportion of corn, this creates a very accessible whiskey.

Here you can find the whole thing described in more detail...