The Vita has a built in PSP 'emulator' which is capable of being exploited to run a full PSP system environment much like the old PSP CFWs. It requires you to be on firmware 3.51 or lower, and you also have to purchase an exploitable PSP game from PSN.
http://www.zload.net/bilder/PSVita/VitaExploitChart.png This is a chart of all exploitable games and their compatible firmwares. You'll also want a guide on setting up either TN-V or ARK, this is good for 3.18 but should be applicable to higher firmwares as well
http://wololo.net/talk/viewtopic.php?f=65&t=39736 But you should do your own reading to be sure.
The PSP emulator mode can run PS1 games, but not with sound. There are seperate PS1 exploits for max compatibility with those, but only work on 3.20 or lower.
There was also a native Vita exploit recently released under the Rejuvenate banner. It exploits the debugger capabilities of the PS Mobile app, so while it doesn't need you to buy a game, it does require you to have the PSM Dev Assistant installed. The window for installing the regular Dev Assistant is already closed, but developers are already working to port all the relevant tools to the Unity version of the app, which is DRM free and so can be freely installed on any <=3.51 Vita.