Area of Effect
From Diablo Wiki
Area of Effect, or AoE is a general term for effects, skills, spells or abilities that will affect an area, and the inhabitants of that area, rather than a single target. Also known as Splash Damage or Line Damage.
Area of Effect spells are sometimes referred to as Area of Denial attacks, since they occupy some portion of the screen and deny access, forcing players to move elsewhere.
AoE In the Game
Many attacks in Diablo III will have the potential to affect more than one target. Some skills will need to be upgraded with points in a mastery skill (the Wizard's Tornado originally produces one tornado, but adding skill points to Epic Storms increases the amount of tornados allowed at once) or used in conjunction with Skill Runes (the Witch Doctor's Skull of Flame used with a multi-strike rune allows the skull to bounce along the ground and explode multiple times) to affect muliple targets.
AoE attacks are best used on multiple targets, especially when they're grouped by some expert crowd control. AoE attacks are usually less effective on single targets, especially ones with a lot of hit points like a boss, though some AoE spells are so powerful that they can work well against anything. Especially if the target moves around a bit, making single-target attacks less effective.
Expect a lot of AoE in D3
The Diablo 3 team is planning on many lots of attacks AoE, either full field magical blasts, or granting multi-target striking to melee and other physical skills. 
- Jay Wilson: It’s because we like them. Diablo II was most fun when you were laying down these massive area-of-effect abilities. Really, every class came down to how it deals with multiple enemies so that has become a big focus for us: a lot of big attacks that hit in an area-of-effect. Or, fast attacks. That’s our other side. If you don’t attack a lot of enemies at once, you attack very quickly.
Most of the known skills and spells in D3, even simple melee attacks like the Barbarian's Cleave will strike multiple enemies if they are in range. This adds a greater element of realism in D3. These types of attacks only hit the targeted monster in D2, even if several enemies were standing virtually on top of each other.
An AoE spell or effect does not need to cause damage. It can give nearby players a buff, give monsters a debuff, or be a combination of the above. Some AoE effects will affect the environment, which in turn effects anyone walking in to that environment.
Diablo II and Diablo III do not have friendly fire for allies, and this includes AoE spells. The game would not be even remotely the same if friendly fire was used. Generally, only positive effects will apply to allies of an AoE caster.
The actual area of an AoE skill can be of any shape or size. Some examples are: circular AoE emanating from the player (Wizard's Frost Nova), a specific area the player chooses (Wizard's Meteor Storm), Line Damage (Wizard's Disintegrate), or jumping from one target to another (Wizard's Electrocute). Out of the lot, the
A "line AoE" damage does not need to be a straight line either. It can be crescent shaped, or swirly, but the main part is usually that it's significantly longer than wide in the area of effect.
Female Wizard using Meteor Storm. Multiple AoEs spell.
Female Wizard using Disintegrate. Line AoE spell.
Female Wizard using Electrocute. multiple target direct damage spell that's "AoE-like".
Besides different shapes, there are also different ways that the AoE effect functions. It's usually a stationary effect, or moving, and when it's moving, it usually consumes itself on obstacles and enemies. The area can also expand or contract, depending on effect.
A few examples:
- A classic Fire Wall in Diablo II or Diablo I will be stationary, and anyone walking in to it will be damaged. The duration is set in the spell cast, as is damage. This is a quite classical shape and functionality to a spell. It's a long shaped cube of fire damage. These are sometimes also called Area of Denial, as it makes the player move away, changing up the gameplay.
- Wizard Frost Nova is an expanding circle (note: not "circular") that will hit anything that is not behind cover from other monsters, or obstacles. It only hits the first few enemies close to the caster. It's another classical shape, and works very different from the Fire Wall.
- Some similar spells are not a true circle AoE, but rather a large array of "bolts" of one form or another. The effect is very similar, but at a little further range, the space between the individual magical projectiles can be so large it misses enemies, wasting that projectile. It's not a real AoE spell, but it has AoE-like functionality.
- Wizard Disintegrate is a line AoE damage that is not consumed when hitting monsters. All monsters in a line will get damaged by the spell.
- Wizard Meteor Storm can be seen as a large AoE damage spell in general, but in reality it's a barrage of smaller AoE damage effects. Each meteor has a small circular AoE damage, and multiple of them will cover a large area.
- Wizard Electrocute is not exactly a real AoE spell either, as it's a multiple target direct damage spell. The spell will hit a designated target, and then a few other randomly selected nearby targets, but not necessarily all targets. It's different than the Witch Doctor spell Skull of Flame combined with multi-strike rune, which work more similar to the Meteor Storm spell, as it has multiple small areas of effect.