History
When Christie's auctioned off a second century Roman glass 20-sided die in 2003, dice aficionados rejoiced over the proof that dice games had been popular for millennia. Later discoveries in Mexico pushed that date back even further when archaeologist Barbara Voorhies discovered strange patterns in clay floors in huts in southern Mexico. Those patterns coincided with accounts of gaming boards from native peoples of the southern United States, offering credence to the idea that humans have been playing with dice for over 5000 years.
Educational
Dice games are used by teachers and educators to teach students valuable math skills. There are several book-and-dice sets available that detail the rules for dozens of educational games, usually grouped by age or grade level appropriateness. "Exploring Math with Polyhedra Dice: Skill building Activities in Problem Solving (Grades 4-8)" by Learning Resources, "Polyhedra Dice Games, Grades K-6" by Don Balka and "Think Fun Math Dice" by Think Fun are just a few of the options available for educators.
Role-playing Games
Perhaps the most iconic use of polyhedra dice is in Dungeons and Dragons and subsequent role-playing games. Dungeons and Dragons uses six different polyhedra dice: the d4 (shorthand for a four-sided die), d6, d8, d10, d12 and d20. Savage Worlds is another role-playing game that uses all six "standard" polyhedra dice. Other role-playing games use a variety of polyhedra dice, notably groups or pools of d6s or d10s. The D20 mechanic, used in Dungeons and Dragons, has even been licensed and is now used by other game publishers as well.
Other Dice Games
Some games use dice exclusively. Games like Yahtzee!, Zombie Dice, Cthulhu Dice, and a long list of others use groups or pools of various polyhedra dice. Games like Dice Poker and Dice Uno use dice in place of playing cards, with the pips on the dice representing the values of the cards. There are also board games that have a dice version, such as Settlers of Catan. For a more comprehensive look at dice games, the odds and strategy, see "Dice Games Properly Explained" by Reiner Knizia.