PPSSPP
PPSSPP is the best, original and only PSP emulator for Android. Not all games run perfectly yet but it runs a lot of games.
No games are included with this download, see below.
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...

DOWNLOAD LINK: https://play.google.com/store/apps/details?id=org.ppsspp.ppsspp