AoE stands for "Area of Effect". It is synonymous with "splash damage".
AoE is used to describe an attack that does damage over an area. AOE weapons are generally more lethal and require less pin-point accuracy, but this is not always the case. They can be used to hit players hiding behind walls or other obstacles out of direct sight. Explosives are the most common AoE weapons, and can be lethal even if not hit directly. Usually, the further away the projectile impact is from the target player, the less damage is received. Therefore, the best use of an AOE weapon is to have the projectile hit as close to the enemy target as possible.