DroidFish Chess

Features:
- Opening books: Internal, polyglot, CTG
- Clocks
- Analyze mode
- Two player mode
- Edit board
- PGN import/export/edit
- FEN/EPD import
- Adjustable playing strength
- One Touch Moves
- Blindfold mode
- Color themes
- Animated moves
- Scid DB support using "Scid on the go"
- Highly configurable
- Third party UCI engines
- Gaviota endgame tablebases
- Free, GPLv3

Source code is available from my web page.

The following permissions are used:

Modify/delete SD card contents
To save games to SD card

Full internet access
To optionally communicate with network chess engines

Prevent phone from sleeping
To optionally prevent screen blanking during game play

Control vibrator
To optionally vibrate after computer makes a move