This page describes the concept of area effect damage and some of the common ways it can be used, which classes excel at it, and even AE soloing.
AE stands for "area effect." Some players choose to instead use "AoE" for "area OF effect" to distinguish it from the common abbreviation of the Mage spell Arcane Explosion which is both abbreviated as AE and is also an AoE spell.
An AE ability is one that affects more than one target, usually any possible target that enters a particular area. This can because there is something about the environment taking place in that area, such as a Blizzard, or perhaps just a character performing an action that can affect multiple targets, such as Whirlwind.
The obvious purpose of AE skills are to be able to affect multiple enemies (or allies) at once. In situations where the player(s) confront a large swarm of weak enemies, such as the onslaught of peons that attack after killing Herod, the party will quickly be overwhelmed by strength in numbers. Although each of the targets dies very quickly, the players can not efficiently kill them each, one at a time. An AE spell would kill them all at the same time, assuming they were in range. (Although some combat abilities are AE, AE is generally associated with spellcasters.)
Because certain dungeons often face scenarios where there are many non-elite targets that must be cleared in one big battle, players often seek to fill group slots for these dungeons with characters more capable of dealing area effect damage. Mages are the best class to fill this role, although warlocks perform very well also. Other DPS classes are not nearly as good, with some classes, like rogues, having nearly no capacity for AE damage. Paladins are known for their ability to tank multiple targets effectively, thanks in part to use of certain AE effects for generating threat, such as Consecration.
A few select classes often choose to level or to farm by grinding on several targets at once. This often requires a special set of talents to be able to do effectively, and is generally only the domain of mages and paladins. It is highly uncommon to see AE soloing before about level 50.
Many MoBs, especially dungeon and raid bosses, will use AE effects to kill off the opposing players. While they are generally tanked by one strong player who can weather the melee attacks, in order to maintain challenge in these encounters, the bosses will assail the party or raid with a barrage of AE effects. Often, a major aspect of raid encounters is in avoiding these attacks by moving to a safe location, or (especially in the case of frontal AE) positioning the boss in such a way that most of the players are safe. In other fights, the AE is unavoidable, and makes some classes superior to others in the fight due to the amount of healing required to keep them alive. Rogues, in particular, generally complain about the amount of point blank effects that they cannot avoid simply because they are strictly a close-range attacker.