The OUYA is capable of emulating many classic consoles, from the Atari 2600 to Nintendo 64.
First, you need to download an emulator from the Discover Store.
Now, after you have an emulator, you need ROMs. Of course, you'll do this the legal way, right? The 100%, honest-to-god, completely, for sure, LEGAL[1] way to get ROMs, because you are a law-abiding citizen. (Heh. Sure, whatever.)
Now that you have your ROMs downloaded, you need to get them on your console. The easiest way to do this is to put them on a USB and then plug the USB into the OUYA. Make sure to extract the .zip if the ROM is in an archive!
Now, you need to open up the emulator. It will ask you to choose a file- choose the ROM you wish to play. If you can't find your USB storage, read the OUYA Filesystem page.

NOTE: OUYA Wiki does NOT endorse nor recommend illegally downloading ROMs off the Internet, not that you care.