What does it mean to be a fan of something? Does it mean anything or it just shows ppl what you like? How do you start chatting with ppl on there? Where do you go to take quizzes?
When you become a fan of something, these things happen:

We shall refer to the "something" as Red, for easy referencing purposes.

1. Red's status updates show up on your homepage.

2. On the right side of the page, people will see "<blank> friends have become a fan of Red" (if a lot of friends are fans of Red).

3. On your profile, in "info" it shows all the things you are a fan of.

To start chatting with people, you have to invite friends. Then, when some of your friends are "online", a box will appear in the bottom right corner that will say "Chat (# of friends who are online)". Click on the box and a list of your online friends will pop up. Click on one of them to start chatting.

To take quizzes, type in "Quizzes" in the search bar but don't click search; rather, click the first item that appears below the search bar. That's where you can take all your quizzes. When your friends take quizzes you want to take too, you can click "Take this Quiz" next to their result.
It usually means, you get updates from it. Like, I'm a "fan" of Nalgene. They make plastic bottles - you know, the kind you carry when you're riding your bike. I get updates whenever they're having a plastic bottle exhibition. (Exciting, huh?) Or whenever they're having a contest you can enter to get a free plastic bottle - except that you have to go to their website to enter.

Or I'm a "fan" of Evernote (piece of software for PCs and Macs and iPhones). I get notices of updates.

Facebook in general means you get in touch with folks you know. You go find them, by name. They find you. You have control over how much information you reveal about yourself to allow people to find you, and you should CAREFULLY research how to control your privacy, so as to know how much you're revealing. Really! Be careful! Don't look at, just the help files. There's other places, not just FB, to read about it. FB gives you lots of control. Use it carefully. Important!

FB will also "suggest" people to be your friends based upon people who are already your friends and who have the suggested people as friends. (Like, Herbert is your friend and Herbert has Juanita as a friend but Juanita isn't your friend yet so FB suggest that you add Juanita as a friend.)

Quizzes will be suggested to you by friends, perhaps. To take the quizzes (which are automated little dealies) you have to reveal your privacy to the programs. Things like, "What kind of super hero would I be?" (and maybe the answer would be Spiderman or Wonderwoman or Hulk).