An Area of Effect is the space in which a Spell, Ability, or effect can apply to a Target.
When using a battlemap, if ANY part of the AOE overlaps the target's model, the Target can be effected by the AOE.
GM Note: You can change this rule to say that the AOE effect has to cover half the target model's base, but this can bring up arguments sometimes. So I rather use the default rule, which is if ANY part of the base is hit, the target can be effected. Also this work both ways for monsters and players, so it a neutral ruling, in my opinion.
AOE is measured from the center of its base. In the case of cube or effects, it is always Length x Width x Height.
Talespire does a great job of measuring this out using the AOE tool for different effects. In talespire you can also put down permanent AOE effects down by right clicking the effect.