An air combo is a kind of combo performed with both characters in the air; it has to be part of a system in a game (typically a combo game), because usually, an air-to-air hit will knock the person getting hit down to the ground.
The air combo varies from game to game, but perhaps one basic form can be thought of in terms of the “Aerial Rave” from the Marvel vs. series (it is offically “air combo” outside of Japan, actually). In that series, certain normal and command moves will launch the opponent high in the air (basically floating); if one jumps immediately after he makes one of them hit, it will change into a super jump that pursues the opponent, and by adding attacks on top of that, it becomes a chain-combo-like sequence constructed of jump attacks.
When this feature was first introduced, it looked unique and pretty cool; it became highly influential, so that today, there are plenty of games that have a very similar-looking air combo system.
Basically, at the end of the air chain combo, there comes a strong attack that does forced knockdown; however, during the Aerial rave, you may instead decide to insert a special move or Hyper Combo, or depending on the character, you may be able to include another jump or two (in the air!). At the edge of the screen, you might also smack them down with the strong attack, then hit them while they are on the ground, then perform a ground chain combo.
The Aerial Rave inspired other kinds of air combos. Despite being made by other companies, these air combos often take the same basic idea: launch the opponent with some special starting move, then jump cancel and pursue them.
In later games, launching attacks often entail + a strong attack, but this, too, is influenced by Marvel vs. Capcom 2.