"Area Effect", commonly abbreviated as AE. Many AE attacks will have a maximum number of mobs within the Area of Effect that will be affected. In EverQuest II this is commonly eight but the number varies from game to game.
While some players use AE and AoE, for Area of Effect, interchangeably, they are not. An AE is a spell that affects all mobs within the Area of Effect, while the Area of Effect is the actual area, diameter or radius.
GTAoE - Ground-Target AoE - Instead of being centered on the caster or on his target, the AoE is fixed to a location and can be avoided by running out of the marked area. For player abilities, this generally means that some kind of targeting reticule appears when you cast the spell so you can tell it where you want it to center. Of course, this varies from game to game.
PTAoE - Player-Target AoE - As a player attack, the effect is centered on the caster and affects all enemies near the caster. When targeted at a player it is attached to the player, not the ground, and will move with the player if he moves. Generally this must be cleansed or the affected player must move away from all others. Some PTAOEs have increased affect if more than one player has them and the circles overlap!
Area of Effect, commonly abbreviated as AoE, is the area, expressed as a diameter or radius, in which the given effect is applied.
This is not the same as AE, which is a spell or effect type. An AE, or Area Effect, spell has an AoE, but not the other way around. In the heat of battle the two abbreviations are commonly confused, and it hardly matters except to us anal-retentive documentation writers.