A subtype of enchantment that can be attached to a specific object, usually a type of permanent. The concept of local enchantments existed since the start of Magic's history, but the Aura subtype was only recently introduced in the Ravnica Block.
The most common target of Auras are creatures and lands. Some auras enhance the creature's stats (ie:Holy Strength), or give it additional abilities (ie:Firebreathing, Phantom Wings). Other auras can be used to reduced an opposing creature's effectiveness, either by preventing it from participating in combat (ie:Pacifism and similar cards), or adding some drawbacks to it. Auras that target lands usually augment its mana generating properties (ie:Wild Growth).
Auras are unique in that they are the only type of permanent card that requires a target to be cast as a spell. In any zone other than the stack, Auras do not have a target. Because of this, you cannot cast an Aura on a creature with shroud, for example, but you can attach an Aura to such a creature if you can manage to put it onto the battlefield without casting it.
Auras can be removed from the objects they are attached to as a state-based action if the enchanted object gains protection from one of the Aura's qualities (such as from enchantments or from the Aura's color) or if the enchanted object stops meeting the enchantment condition (say, if an Aura with "enchant tapped creature" is enchanting a creature that becomes untapped.) In both cases, a second state-based action then moves the Aura to its owner's graveyard.