IMPORTANT: There is also a free version of PPSSPP, with identical functionality: 
https://play.google.com/store/apps/details?id=org.ppsspp.ppsspp

Buy PPSSPP Gold only to support further development of PPSSPP. Feel free to download the free version linked above if you want to.

PPSSPP is a PSP emulator for Android and other platforms. It is being developed at a rapid speed and compatibility is increasing quickly, but there are many games it doesn't run yet and it doesn't run fast on Android yet.

Please go to 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.

PSP is a trademark of Sony, Inc.

IMPORTANT: Please do not give bad reviews due to low compatibility. It takes time.

What's in this version : (Updated : Oct 12, 2015)
mainly fixes a number of crashes in 1.1, see the website.
Warning: Savestates may not be compatible! Make real saves before you upgrade.
1.1:
Support for ARM64 for improved performance on new devices.
Support Android TV, like nVidia Shield TV
Performance improvements, and bug fixes, music fixes
GPU fixes like correct depth rounding, fixing text in Phantasy Star
Support savestates for homebrew
Integrated "Homebrew Store"
Minor AdHoc multiplayer improvements.
Required Android O/S : 2.2+