Out of Competition

RetroArch

This is where you start your journey. Often called the mother of all emulators on Android, RetroArch actually incorporates different cores of multiple emulators. At this moment, RetroArch can provide emulation of over 14 different systems plus Doom and Quake game engine support. Best of all? RetroArch is free and will stay free forever. You can find more info by visiting the developers website.

Sony

Sony Playstation (PSX, PSONE)

Read more about PSX emulation

FPse for Android

FPse for Android is the fastest and most compatible PSone emulator for handheld devices.(also known as PSX or Playstation 1)More than 200.000 active users and rated up to 4,5 stars on google play! Some of the newest additions include OpenGL HD Rendering and experimental LAN multiplayer mode.

ePSXe for Android

ePSXe for Android is a port from the famous ePSXe for PC. ePSXe provides very high compatibility and good speed. It is designed for smartphones and tablets, (for 1-4 players) including a fun 2 players option for tablets using split screen mode

Sony Play Station Portable (PSP)

PPSSPP

While still in early stages of development, PPSSPP can already play number of games. It requires moderately powerful device and lots of patience. If you like to support further development,  you are encouraged to purchase the Gold version of this emulator.