PPSSPP 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. A free version is also available, you may want to try it first.
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.
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...