PPSSPP - PSP emulator is the best, original and only 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. 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
GTA (slow on mobile)
and many more...