Although Facebook said you have to wait for an invite either through facebook.com or from one of your friends who have some invites to be sent to his/her friends.
But If you are eager and want the invite as early as possible, Facebook has set up a page where you can request the social networking website to send an invite to you. 

Once you are on the page, just hit “Request” at the bottom right of the box, you can just wait as according to the Facebook, you will be receiving an invite soon.

Also, if you are in desperate need of an invite, there are some folks on ebay selling the invite for around $0.99, not sure if anybody has received any invite yet from the auction, but you can give it a try.