"Root" is basically the UNIX/Linux/etc term for having administrator rights. It means you can do things that would normally be off-limits for safety and security reasons. If you aren't trying to do anything risky, then there's no risk, but it means that the barrier keeping you from breaking things is your own common sense rather than a built-in system policy.

The best tool to get root access is "Root My OUYA" which you can find over on the XDA forums:
http://forum.xda-developers.com/show....php?t=2387507

Google Play is different from the Amazon store because it needs to run as a system application and keep several processes running in the background. That's why you need to root your system to install it. It also means there could be some very slight performance degradation in some games due to the background processes. Or you may get some error notifications when OUYA closes Google Play's background processes to free up resources. I don't notice any performance issues in games myself, but the possibility is there in theory.