As I understand it, an "air" is another word for a melody. But in folk music it is (as someone has said up there) often used for those tunes that are not easily categorised as dance tunes (jig, reel, hornpipe etc), and therefore tends to denote the slower tunes and song melodies. People often call a slow tune with no discernible dance rhythm an "air" or "slow air" (particularly in Irish and Scottish music), and as it's not being played as a dance tune it may be played more loosely and lyrically.

Not all 3/4 tunes are waltzes. Airs in 3/4 were around for centuries before the waltz came into being; some of these may happen to suit the waltz, but others do not. There are other 3/4 dances (eg mazurka, bourree, minuet) that have different stresses from that of a waltz, and there are also many old song-tunes and free-standing "airs" in 3/4 that are not really waltzes. The tune referred to as "Michael Turner's Waltz", for example, predates the waltz and is more like a minuet in its stress pattern (nor is it Micheal Turner's, but that's another issue).