Auras come in many forms. A list can be found here.
Most auras cling to or surround the character's body. Most auras will damage opponents who make close-combat attacks (Hand-to-hand, Melee, Touch Attacks and Pets), although some auras - including Clockwork Cloud, Acid Blood, Dark Trick, and Harbinger of Judgement - can damage other attackers, such as from ranged weapons and spells.
Aura Spells are a special type of aura and function offensively as well as defensively. They can be cast from memory or Spellgems, and deal supplemental damage to a target each time the caster successfully strikes them in close-combat (or, with Arcane Marksman, charges a firearm attack). They last anywhere from 5 to 10 status ticks, based on Spell Affinity, and their duration can be stacked. Remember that being attacked by pets will not reduce their duration. Autohit spells are considered ranged attacks and do not receive the bonus damage from an active aura.
A character cannot have multiple aura spells active - casting an aura spell of one time dispels any other type of aura spell active on the character. It is possible for a character to have both a spell aura and an innate aura active (e.g. a Lich with the Aura of the Crypt skill casting Static Charge), but in such cases only the aura with the highest damage (before applying soak, resistance, etc) will actually function defensively.