Is it like a contest in another site?

After a fashion. Google "NaNoWriMo" and you'll find the link. There's also the Young Writer's Program, which is basically the same only they don't zap 12-year-olds with accounts.

Is it a contest at all? Are we judged and given prizes, or is it more of a thing that someone on Cricket thought up for fun?

After a fashion. You're not judged, all you have to do is copy/paste your novel (or an encrypted version of your novel) into the wordcount validator on the official site to become a "winner". Your prize is a 50,000 word rough draft of a novel, a pretty printable certificate, some web banners, and a warm fuzzy feeling inside. And your username on a list somewhere.

Do you HAVE to do a novel in a month, like, what if an emergency came up?

You don't have to. It's not as if someone's going to show up at your door with a baseball bat if you don't finish. It's mostly for fun. Therefore if an emergency happens, like, I dunno, your house burned down or a family member died, obviously that's more important than writing a novel.

Are there any rules?

Yes, you can find the official set on the official site, but basically the rules are as follows:

1. You cannot start before 12:01 on November 1st at the earliest. You must stop writing at 11:59 November 30th at the latest.

2. Your goal is 50,000 words (or higher). That's ~1,667 words per day (or higher). ((For example, my goal is 100,000 so I have to write 3,334 words a day).

3. No, you cannot write one word and copy/paste it 50,000 times. While NaNoers encourage various methods of cheating (i.e., extensive quotes, song lyrics, even binary in some cases), doing that's just silly.

4. Please don't give yourself carpal tunnel and then sue the Office of Letters and Light (name of company that runs the NaNo site). That's just rude.

And that's about it, really.

Do you have to sign up on a different site?

If you want the winner's certificate and web badges, yes. If you don't care about those, unofficial NaNo's get just as much respect as official NaNo's, and you still get the rough draft. So you only have to sign up on the NaNo site if you want to.

Do you have to sign up in a thread, or do you just start? 

See previous. You can just start whenever (but write like crazy to catch up) and you can post exerpts if you want.

Hope this helps, and good luck.