They're torrent trackers, like That Penguin Bar, except they have an account system. Normally,this would not be a big deal. But on most private trackers, it is very difficult to get and maintain an account. You have to be invited by an existing user, probably even one who's an experienced member of the site. In order to get these people to invite you, you need ratio proofs, proof that you have kept a good ratio on another similar site. That brings me to the other bit. In order to maintain a private tracker account, you must seed. The general rule is to upload at least as much as you download. If you don't, they'll kick you out.
However, because this stringent system keeps bad users out, and because upload credit is so prized here, private trackers are the first torrent sites to get new releases. Additionally, the users are so good that torrents are rarely without seeds, and the seeds they usually have are really, really fast.
So in short, they're exclusive torrent trackers that are difficult to get onto and use, but have superior content. I can't give any examples.
I joined one devoted to anime. They're very good because whenever a torrent is low on seeds, they either start up a seedbox or give incentives for people to start seeding it more. That way they maintain their entire library. They also filter out double uploads.
Like what Midna said, it can be rather difficult to maintain a good ratio on private trackers. I have been a member a multiple, and there has only been one I managed to keep myself afloat long enough to get a foothold. I'm a well respected member of said site and the community is great. Been there for 3 and a half years now. So, in short, if you can sustain a good ratio and become an active part of the community, you will reap the rewards.
Being closed to anyone and everyone that wants to sign up is precisely what makes it a private tracker.
Yes they're absolutely awesome, and I'm closing the thread before you can ask for an invite to one.