Regular Backgammon
The Regular game type implements the regular rules of the game. In Regular Backgammon, you can pile as many tokens as you want on a point and you must play both dice if possible. See The Basics for more complete rules.
Old English
The Old English variant is very popular in England. There are 2 major differences with the Regular Backgammon rules. The biggest difference is that you're only allowed to place a maximum of 5 tokens on each point. In Old English, you are also not required to play both dice if possible.
Nackgammon
Nackgammon is played with the same rules as Regular Backgammon. The only difference is in the initial position of the tokens. Each player starts with 2 additional back tokens. As a result, Nackgammon games tend to last longer, requiring more emphasis on position and back games.
Acey Deucey
Acey Deucey is played with the same rules as Regular Backgammon. The tokens start out the game off the board and are entered onto the opponent's inner board as if they were captured. When a player rolls a 1 and a 2 (Acey Deucey) they first play that roll, then they get to pick a double of their choice. Once the double is fully played, the player gets to roll again and keep playing. If the new roll is another 1 and 2, you keep going, naming another double and rolling again. If you are unable to use part of the roll, the rest of the turn is forfeited and play moves to the opponent.

The winner earns 1 point for each of the opponent's tokens still on the board. Gammons and Backgammons do not count for extra.