The “air dash” (or “midair dash”, “ærial dash”, etc.) system allows a motion much like a dash that is performed in midair. In its typical incarnation, it advances forward (perhaps sometimes backward) a certain distance quickly, deviating from the expected trajectory.
The air dash originated in the Darkstalkers series (with Zabel), then found its way into the Marvel vs. series (with several characters); because of the possibilities it opened, it has been found in many games since. For most recent purposes, it has been mostly a feature of 2D combo games like Melty Blood and Guilty Gear (along with double jumps). Also, much like triangle jumps, some games only have an air dash on some characters. It is not very common at all in fighting games that aren’t combo games.
For the most part, the air dash is different in function than that of the ground. One travels a fixed distance forward or backward, then descends. In a few cases (but not most), you can continue the dash by keeping the stick in that direction (like with Neco Arc in Melty Blood).
In some cases, you may air dash in all eight directions, though this is more common with certain kinds of special moves (refer to Choi Beon-Gae or Ruby Heart), since it can be quite tricky.
The Marvel vs. series (as well as Tatsunoko vs. Capcom) allows you to hit buttons in the same row simultaneously to perform any dash, making it simple to perform an air dash while ascending.