Well you would learn exercises that are used in competition obedience.
For instance in your basic class you probably learned heal, sit, down stay, and a recall.
In advanced you will focus more on healing off lead and getting precision. ie, a nice straight sit instead of slumped onto one hip.
You may also be introduced to formal retrieval work.

Is the class you might be taking being taught by someone who has competed before? If so they will be able to help you get started. I hope you do decide to try going to trials. It is a lot of fun, I love it. Good luck.

Here's a simple break down of AKC obedience:
http://www.akc.org/dic/events/obedie...ng_started.cfm

And also the rules for Novice obedience.
http://www.akc.org/registration/rules/RO2999.cfm?page=4