A pun itself, often thought to be the lowest form of wit, is itself a play on words.  Authors use them to add meaning or complexity to their works by having their words take on more than one meaning.  Much of the humor derived from The Importance of Being Earnest is from Wilde's use of puns.

The title of the play is in itself a pun.  As the play concludes Jack says

On the contrary, Aunt Augusta, I've now realised for the first time in my life the vital Importance of Being Earnest. (III.180-181)

The pun is that his names is Earnest which also means to be truthful.  While he's been lying (and thus not being earnest) for the course of the play, he now knows his real name and the importance of being truthful.

The pun also comes when we look at Gwendolyn's desire to marry someone named Earnest.  She doesn't care if he holds earnest qualities or not, as long as his name is Earnest.

In Act III Lady Bracknall says about Jack's origins that she "whose origin was a Terminus" she again makes a pun.  Jack, an orphan, can only race his origins back to bein found at a railroad station, also called a terminal. The first stop on a railway is considered its origin and it's final stop is its terminus.