As noted below I have Aart Bik's chess app and it's good for playing blitz, but after some research I downloaded DroidFish to use for those times I want to do some serious analysis.
DroidFish has to be one of the strongest free chess programs available because it uses the Stockfish 6 engine.
You can adjust SF's playing strength from 0 to 100% but I have no idea what its estimated playing strength might be if it's set to, say, 50 per cent. You will just have to experiment.
DroidFish allows you to do the usual stuff you find in other full-fleged chess programs: show thinking lines, adjust thinking time, enable a time increment added after each move, display book hints, adjust pieces, square colors and font sizes, save games into pgn files and include comments. You can also load previously saved games.
DroidFish has an opening book, but it can also can use external opening books which can be stored in the internal memory as well as on a card. Position set up is also available just in case you want to analyze a particular position.
Complete Features List:
- Opening books: Internal, polyglot, CTG
- 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
- Configurable UCI engine options
- Support for "open exchange" engine apps, such as texel and komodo 8.
- Gaviota endgame tablebases
- Syzygy endgame tablebases
- Free, GPLv3
It does lack a few important features like the ability to use multiple engines for analysis and automatic annotation, but for a free Android app, it looks to be the best available.