An air/aire is the melody of a song, in ITM usually of a an old song sung in the old style (sean-nós). Get some recordings of sean-nós singing to get an idea of timing. Many instrumentalists unfamiliar with the sung versions play the tunes far too slowly, so it’s best to understand how the sung version goes.
Not all the old tunes survive as sung tunes, just as melodies. Listen to traditional players to get an idea of the speed for these tunes.
An air is a term for the melody of a song, in usage since at least the victorians and prolly earlier. Could be a hymn, a carol, a Robbie Burns song like Ye Banks and Braes, one of Thom Moore’s* songs, or something like Scotland the Brave.
A slow air, otoh, is the melody attached to a song in sean nos style; whether sung or played instrumentally by a soloist.
*Anglo-Irish poet & lyricist remembered for sentimental song lyrics in english like The Minstrel Boy, etc.
An orthographical point: the term, as used conventionally, normally, or usually in Irish traditional music and in any modern context associated with the uilleann pipes, is spelled ‘air’, as in ‘slow air’ or ‘song air’. There is no ‘e’ on the end of ‘air’, unless one deliberately wishes to be twee.