Backgammon Variants
There are many variants to backgammon, including those found here. Here are some of my favorites.

Dice

Trigammon (Gregg Cattanach): Roll 3 dice, and move all 3. Doubles do not count extra, but for triples you get to move 6 of the number thrown.

Modified Veto Trigammon (James Eibisch): Roll 3 dice. If you are not on the bar, your opponent chooses which 2 form your roll. If you are on the bar, you decide which 2 form your roll.

Choice Trigammon: Roll 3 dice, and you choose which 2 form your roll.

Quadgammon: Roll 4 dice, but ignore any that appear more than once.

Fairgammon: Roll 4 dice, your opponent chooses 1 die to remove, and then you choose 2 of the 3 remaining as your roll.

Multiplegammon: Roll n sets of colored dice, and choose which color to play.

Vackgammon: Roll 1 die into a V-shaped structure, so that 2 sides are up, and the player plays those dice. This means there are no doubles or 7's.

Single Die: Roll 1 die, and move either 6-5, 4-3, 2-2, or 1-1. Doubles From the Bar: When you have checkers on the bar, you only roll 1 die, and the other is assumed to be the same. 2 or 3: Every turn, the players have the option of throwing 2 or 3 dice. If they throw 3, any dice that repeat are ignored.

Different Dice

Skewed Dice (Stein Kulseth): Players use a 4-sided die and an 8-sided die.

Eskgammon (Peter Bäckgren): Players use a 6-sided die and an 10-sided die.

No Sixes (Douglas Zare): Players essentially use two 5-sided dice, since 6's are re-rolled.

4-Sided Dice: Players use three 4-sided dice. Doubles count as triples, and triples count as pents. So rolling 1-3-3 lets you move 1, 3, 3, 3, and rolling 2-2-2 lets you move 2, 2, 2, 2, 2.

Duodecagammon (David Moeser): Players use one 12-sided die to roll, and can break any roll 2–12 into any 2 moves or 3 equal moves with that total. So a roll of 9 could be played as 8+1, 7+2, 6+3, 5+4, or 3+3+3.

No Dice

Gabgammon: Every roll is assumed to be 4-3.

No Chance Backgammon (Naim Çagman): Each player rolls one die, and the player with the higher number plays that roll. From now on, each player now changes his die to something different from what his die was and different from what the other player's die is, and plays that roll. Doubles never occur.

Over and Over: Each player rolls one die, and the players take turns moving that roll over and over until a checker is hit. Then the next player rolls until a non-double occurs and that player is able to enter a checker from the bar. That roll is used repeatedly until the next checker is hit.

Domino Backgammon: The dominoes from a double-6 domino set are used, with the blanks removed. Three doubles are given to each player randomly, and the non-doubles are divided 8-7 randomly, with the first player getting 8. A player chooses a domino to play each turn with their men, and turns it over so that it is not used again. After 15 moves, the non-double dominoes are switched and turned over again for the next 15 moves, and so on. Each double can only be used once per game, and only the smallest double owned but not yet used. When a player uses a double, a non-double is also turned over to maintain parity.

Choice: Each player secretly chooses a number 1-6 to be part of each roll.

Modify Dice

Decrement: Players must subtract 1 from one of their dice. So rolling 4-3 means a player moves either 4-2 or double 3's.

Deuces Wild (Walter Trice): 2's are wild, and can be used as any number, even to make doubles.

Double Ones: Any roll of a 1 is moved twice. A roll of 1-1 gives an infinite number of 1's. This is an automatic win for a player in a non-contact situation.

No Ones: 1's count as 7's.

Rock-Scissors-Paper: Each turn, the player rolls one die. If it is a 5 or 6, he can move any roll he wants. If he rolls 1 or 2, his opponent decides what roll he should play. If he rolls 3 or 4, he rolls normally.

Look Ahead (Stephen Turner): After moving their checkers, the player rolls their next roll.

Look Further Ahead: Before moving their checkers, the player rolls their next roll.

Secret Further Look Ahead: Before moving their checkers, the player rolls their next roll, but does not show the other player.

Bluffgammon: Before the doubling decision, a player may see his roll but his opponent may not.

Re-Rolling

Roll-Over (Edward Collins): Once per game, you can choose to re-roll your dice. And once per game, you can choose to force your opponent to re-roll their dice.

Offensive Roll-Over: Every roll, you have the choice of option of re-rolling your first roll of the dice.

Defensive Roll-Over (Fredrik Dahl): Every roll, your opponent can force you to re-roll your first roll of the dice.

Offensive Half Roll-Over: Every roll, you have the option of re-rolling one of your dice.

Defensive Half Roll-Over (Fredrik Dahl): Every roll, your opponent can force you to re-roll one of your dice.

Joker Cube (Joe Russel): There is an additional joker cube that starts in the center. At first, either player can use it to force their opponent to re-roll once, and possession of the joker cube goes to the other person. After that, only the player in possession of the joker cube can force their opponent to re-roll.

Not the Same: Players repeatedly re-roll any dice that were part of their last roll.

Blots

Blocking: Hitting is illegal, so blots count as points.

Immunity: Hitting the only blot of that color is illegal.

Guards: Hitting is illegal, unless hit by multiple checkers on the same turn.

Immune: Hilling is illegal in your inner board.

Sudden Death (Fredrik Dahl): A player who hits a blot wins.

Hit and Die (Cliff Kilt): A player forced to hit a blot loses.

Pinned: Blots that are hit become pinned, unable to move until the checker above it is moved. The pinning checker can itself be pinned, unless a point of two consecutive checkers of the same color is made.

Forced Hit: Your opponent can force you to hit a blot, though you retain the choice of which to hit or how many.

Starting Configurations

Nackgammon (Nack Ballard): Both players start with with the usual configuration, except that 1 checker each from the 6-point and 13-point start on their 23-point.

Quickgammon: All the checkers start one pip closer to home.

960: There are 960 equally likely possible starting configurations. See this page.

Long Run (Bill Hickey): Both players start with 5 checkers on their 24-point, 4 checkers on their 19-point, 4 checkers on their 13-point, and 2 checkers on their 8-point.

Longgammon (Micheel Strato): Both players start with all 15 checkers on their 24-point.

Others

Exact Bearoff (Chris Moellering): You must roll the exact number when bearing off.

Misère (Stein Kulseth): The player that bears their checkers off first loses.

Damage (Gregg Cattanach): Both players move their opponent's pieces for the first 5 moves of the game, then they play their own. No doubling cube actions allowed during the first 5 moves.

Backwards (Colin Bell): Backwards moves are possible, and even forced if moving forward is impossible.

Chase: Both players are moving clockwise around the board, bearing off on their left.

Poof (Walter Trice): Players must use their lowest roll first, and if they cannot, they do not get the other half of their roll.

Madgammon (Vadim Madgazin): Any subset of a point can be moved as if they were a single checker.

Veto: Your opponent can veto one possible move per turn, assuming there is more than one legal move.

Taxicab: Moves of checkers count not only spaces moved forward but also change in height of points. For example, rolling 3-1 on your opening roll cannot make your 5-point, but can make your 7-point from your 8-point (2 down and 1 over, and 1 over).

Me Too: After your opponent has rolled and moved, you can say "me too" and use his roll instead of rolling.

You Too: After you roll and move, you can say "you too" and force your opponent to have the same roll instead of him rolling. Small Board: On each quadrant of the board, only 5 of the 6 points may contain checkers. This means you can close your opponent out with only 5 points!

Lake Wobegon: Rolls are computer-generated, and are randomly chosen from the 18 different "better than average" rolls.

4-a-Bonus: If you did not roll doubles, then playing any die that gives exactly 4 men on a point lets you re-roll that die for additional movement.

You're the 1: Your opponent moves all your 1's, before you play the rest of your roll. This usually makes 1-1 a VERY bad roll.

Stack 'Em Up!: The winner of the game is not the player who bears off checkers first, but the player who had the most checkers on the 6-point at some time in the game.