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
GTA (slow on mobile)
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.
Please note that ARMv6-based devices may have various kinds of issues, and more modern devices are recommended.
* PSP is a trademark of Sony, Inc.