Burlesque is a comic style that works in one of two ways: you can either elevate something lowly and ridiculous (high burlesque) or trivialize something lofty and important (low burlesque).
Burlesque makes audiences laugh because of the difference between the content and the form (the style and the substance).
An example of high burlesque is Alexander Pope's "The Rape of the Lock", a poem that takes something seemingly trivial (a lady getting a lock of her hair cut off) and turns it all into a big ol' hullabaloo. You'll have to read it to find out just what that hullabaloo was. For low burlesque, check out Samuel Butler's Hudibras.
In modern usage, burlesque can also mean a kind of striptease or stage entertainment.