PPSSPP is the best, original and only PSP emulator for android It runs a lot of game , but depending on the power of your device all may not run at full speed. Buy PPSSPP Gold to support the development.
Please note that ARMv6-based devices may have various kinds of issues, and more modern devices are recommended.
Some examples of compatible games (not all will run full speed on all devices):
Persona 2, Persona 3
Little Big Planet
Burnout Legends, Burnout Dominator
Final Fantasy : Crisis Core
Monster Hunter 2 Unite
Soul Calibur
GTA (slow on mobile)
Daxter
Lumines
Worms
and many more...
See http://www.ppsspp.org for more information
No games are included with this download you will have to dump your own 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.
* PSP is a trademark of Sony, Inc.