An air throw is a move that can grab your opponent in midair and throw them when your and their air hitboxes are near each other. If it has a simple normal throw-like command, it can even outclass special moves. It trumps air blocking and parrying.
In Street Fighter II, only Chun-Li, Guile, and Balrog could use air throws; in most later games, only parts of the character cast can use them. There are plenty of exceptions: in the Capcom vs. series, Pocket Fighter, Street Fighter Alpha 3, some of the Guilty Gear series, the Arcana Heart series, and so on, all characters may use air throws.
Also, most of the time, grapplers (especially the more heavyset ones) don’t have it; rather, it tends to go with more nimble characters, like those who use ninjutsu or Chinese martial arts, or smaller people.
In the Melty Blood games (as well as some similarly-styled ones), they can be included in combos, so they generally go best to wrap up an air combo.