Fury is a stat that the Barbarian builds up during combat. Successfully hitting enemies fills up the Barbarian's Fury bulb, enabling him to use his more powerful skills, most of which cost some Fury to use. When the Barbarian is not fighting, or not hitting enemies, his Fury steadily fades away, until it drops to nothing several seconds after a battle.
Fury changes the Barbarian's play style, making it impossible for him to use his most powerful abilities at the start of a fight. He must warm up a bit, building up his Fury with basic attacks, and only unleashing his devastating skills when the Fury bulb has filled up enough to allow him to expend the energy in violence.
Fury is modified by a number of skills, some of which cause it to fill more quickly and drain more slowly.
Few particulars are yet known about Fury. How is the maximum value determined? Is there equipment that will boost the total, or cause it to drain more slowly or fill more rapidly? Are there potions that will fill the Fury bulb? Does the Barbarian gain Fury when he or a party member collects a mana globe?
Lots of active skills don't currently list a Fury cost; are those skills going to be free to cast in the final game, or is their non-cost status more about the game's documentation being far from complete?
Other questions abound, and until the D3 team speaks more about this aspect of the Barbarian, we will continue to have many more questions than answers.