This was a huge mystery to me when I started submitting. I had no idea what came after, 'we'd like to offer you representation.' I just read an article in Writer's Digest that reminded me of the whole terrifyingly wonderful process. Having now been through a portion of it, I shall demystify it a bit for you.
First, here's one myth I'll squash: Agents expect you to sign a contract with them right away. Wrong! Some will, some won't. Especially if you're a first time author they may not want to tie themselves to you until they know they can sell you. That way if things don't work out they aren't saddled with a book they can't sell or a client they can't work with. Don't panic if they don't expect you to sign a contract immediately. If you did your job right to begin with and researched them, checked their clients, made sure they were a member of AAR (there are reputable one's who aren't a member, just tread those waters cautiously), then you're probably safe. If you have any doubts, check again.
The biggest slab of advice I can give on the subject is something you've probably heard a million times, but deserves reiterating. Reputable agents don't charge reading fees! Of any kind! It's a big red flag if they do. The only expense you should have is sending them a hard copy of your manuscript, or any other mailed correspondences. I don't care what they say they can offer you or who they know, if they charge a reading fee, they're probably bogus.
When you get to the stage of signing a contract with them read it carefully. Don't make a knee-jerk decision and swoon over them immediately. Remember, finding an agent is finding a long term business partner. You want to like them and you want them to like you but don't worry, you don't have to be the best of buddies. Make sure the contract is to represent your book, or series (be specific about the number therein!), not to represent you. You don't want to sign with them for all eternity (or a ridiculous or unknown number of books) and later come to realize you don't work well together. The same thing goes with the publisher. If you don't think this is stated clearly enough in the contract ask the agent about it. They're people too, they'll listen and address your concerns. If they don't, then there's another red flag!
Getting an agent is an exhilarating process that shouldn't be painful. You've waited a long time to hear those words. Keep your head about you, don't get starstruck, and you increasing your chances of it being a wonderful partnership.
Tips from Writer's Digest: http://www.writersdigest.com/article/writer-agent-negotiations/
Advice on a problem we all hope to have, choosing between agents, by agent Caren Johnson: