The basic system of resolving conflicts in the World of Darkness and Age of Sorrows role-playing games involves the use of ten-sided dice, usually three to ten dice, though Exalted is notable (or infamous) for requiring upwards of twenty or thirty dice to be rolled at a time.
The number of dice to roll is determined by some factor, usually a combination of Attribute plus Ability, based on the player character's statistics. For example, a "Dexterity + Melee" roll for a character with a Dexterity score of 4 and a Melee score of 3 involves 7 dice.
In the World of Darkness game-line, the Storyteller dictates a "target number"; this is the number a player wishes to roll at or above on his dice. The Exalted system uses a flat target number of 7 for almost all rolls.
The player then rolls his allotted number of dice, counting the number of dice that came up at or above his target number. This total is the number of "successes" achieved for the action the player rolled for. For example, a roll of five dice with 7, 7, 3, 2, and 10 would be a total of 3 successes, assuming a target number of 7.
The World of Darkness games offered a "specialty" rule, where a result of 10 on any die counted as two successes when the roll pertained to some specialty the character had trained in or was adept at. In Exalted, 10s automatically count as two successes for non-mortal characters (for example, Exalted, spirits, Fair Folk, and other magical beings).
In the World of Darkness games, each 1 rolled subtracts an effective success. In Exalted, 1s do not count unless the player also rolled no successes. In either case, the result is a "botch", or a spectacularly bad failure at the task being attempted.
Ordinarily, only a single success is required to achieve the player's goal. In many cases, particularly trying or challenging tasks are assigned a difficulty, a minimum number of successes needed to succeed. For example, convincing a suspicious guard of your good intentions may be a "difficulty 3" roll; it is made harder because of the guard's wary nature, and the player must roll at least 3 successes to succeed at all.
More complicated variations on this basic idea are introduced in each game-book's rules, such as opposed rolls.
