PPSSPP is the original and best PSP emulator for Android. It runs a lot of games, but depending on the power of your device all may not run at full speed.
No games are included with this download. Use your own real PSP games and turn them into .ISO or .CSO files, or simply play free homebrew games, which are available online. Put those in /PSP/GAME on your SD card / USB storage.
This is the free version. If you want to support future development, please download PPSSPP Gold instead!
See http://www.ppsspp.org for more information, and see the forums for game compatibility information.