PTR stands for public test realm.
The PTR comes out from time to time to test new patch content. It's kind of like a mini Beta test. Blizzard uses it as an opportunity to test new balance issues, fine tune new bosses to appropriate levels, find bugs, and gather input from the community.
Players play on the PTR for a number of reasons. One of the biggest reasons is the opportunity to learn new content before it goes live (to the normal servers). The "elite" guilds go so that they can learn encounters to be better prepared to get world first, US/EU first, or Server first kills. All PTR players also get to have a more direct affect on the future of the game by giving feedback on changes.
As for 3.1, it is coming out soon. Right now it is finishing up on the PTR. When it becomes available you will need it. However, the blizzard downloader will download it and install it automatically for you.
If you're worried about the background downloader, let it run as much as possible. The patch is BIG (1GB+) so having to do it all at once when you want to play will be very time comsuming.
Best of luck and for the HORDE!!!
=D
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