An aerial attack is any attack performed in the air that's not a special move.
There are five standard aerial attacks, performed by pressing the attack button in any 4 directions or no direction at all while in the air - The neutral aerial, the forward aerial, the back aerial, the up aerial, and the down aerial. 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 aerial hits right in front. Depending on the character it can be a multi-hit attack, a strong finisher or a spacing tool.
Up aerial hits straight up or in an arc, mostly used for juggling or as a vertical finisher.
Back aerial hits in the opposite direction the character is facing. Some have good range making them useful for spacing. Performing a RAR makes approaching with them possible.
Some characters (Link and Samus) can also use their tether recoveries as an attack while midair, this is also known as a grab aerial or zair. These moves have long range, come out fast and have low landing lag, making them effective for spacing and poking.