Below is a brief list of just some of the popular backgammon games played around the world. There seem to be endless variants, all with distinct and unique challenges. Some of these games are still being researched, so forgive any holes in the information.
Chouette (game for 3 or 4 players) - A captain and team play against the box
Dutch Backgammon - Both players start with all their checkers off the board
Turkish Backgammon (Moultezim) - This game is similar to Narde (Russian) and Fevga (Greek variant). There is no hitting and the players move in the same direction around the board. One checker also controls a point forming a block.
Greek Backgammon (Plakoto) - Rather than hit, opponent's checkers are pinned in position. The same game is played in Bulgaria (called Tapa).
Gul Bara - Similar game to Moultezim in that one checker controls a point. However, doubles are very powerful in Gul Bara Gioul - Middle Eastern variant with setup and movement like Plakoto, a single checker on a point forms a block as in Moultezim, and doubles are very powerful as in Gul Bara
American Acey Deucey - Popular version of the Dutch game played by the US armed Forces since World War One
European Acey Deucey - Doubles are played on both sides of the dice
French - Both players start with all their checkers off the board and doubles are played on both sides of the dice
Russian Backgammon - A race style game where both players move in the same direction