There are five standard aerial attacks, performed by pressing the attack button in any four directions or no direction at all while in the air. Each attack is usually set so that the hitbox of the attack will hit in the general direction of that aerial attack.
Neutral aerials, for the most part, are fast attacks, hitting either in front or around the character. Some have lingering hitboxes and are referred as sex kicks.
Forward aerials hit right in front. Depending on the character it can be a multi-hit attack, a strong finisher or a spacing tool.
Up aerials hit straight up or in an arc, mostly used for juggling or as a vertical finisher.
Back aerials hit in the opposite direction the character is facing. Some have good range making them useful for spacing. Performing a reverse aerial rush makes approaching with them possible.