I stand corrected!
From my limited research, all I’d found were some place names http://en.wikipedia.org/wiki/Aire which arguably come from the French.
I hadn’t come across the old spellings.
Nonetheless I stand by my assertion that it is only used by people to refer to slow airs if they are either 1) being careless with spelling or 2) clueless. (Or ironic, or as Kieran says, twee https://forums.chiffandfipple.com/t/what-is-an-air/59157/5)
Agreed! 
This topic has inspired me to to exclusively spell it “Aire.”
To aire is human.
naw, I hadn’t mucked stalls yet 
I don’t ride, I drive!
there should be a joke or two in there somewhere…
Bey shur ta aire thy tyres ere ye goe.
I need to get solid ones…someday
Sorry you asked, Mason? 
English had no standardised spellings until the latter 17th century or so. Before then, people wrote what they heard, and reading aloud or moving their lips was how everyone–not merely children & morons–read.
Reading visually (recognising words as shapes) which every educated person does these days wasn’t how the vast majority of (if not all ) people read. It requires standard spellings, which didn’t exist. There are many medieval manuscripts in which the same word can have three different spellings on the same page. The text of Sir Gawain and the Green Knight, for instance, can have multiple spellings of Gawain within a few lines. The elipsis below represents one ‘fit’ (stanza/verse/‘unit’ of poetry) that I left out, but the bolded words are both the name ‘Gawain’.
After þe sesoun of somer wyth þe soft wyndez
Quen Zeferus syflez hymself on sedez and erbez,
Wela wynne is þe wort þat waxes þeroute,
When þe donkande dewe dropez of þe leuez,
To bide a blysful blusch of þe bry3t sunne.
Bot þen hy3es heruest, and hardenes hym sone,
Warnez hym for þe wynter to wax ful rype;
He dryues wyth dro3t þe dust for to ryse,
Fro þe face of þe folde to fly3e ful hy3e;
Wroþe wynde of þe welkyn wrastelez with þe sunne,
Þe leuez lancen fro þe lynde and ly3ten on þe grounde,
And al grayes þe gres þat grene watz ere;
Þenne al rypez and rotez þat ros vpon fyrst,
And þus 3irnez þe 3ere in 3isterdayez mony,
And wynter wyndez a3ayn, as þe worlde askez,
no fage,
Til Me3elmas mone
Wat3 cumen wyth wynter wage;
Þen þenkkez > Gawan > ful sone
Of his anious uyage.[…]
Sir Doddinaual de Sauage, þe duk of Clarence,
Launcelot, and Lyonel, and Lucan þe gode,
Sir Boos, and Sir Byduer, big men boþe,
And mony oþer menskful, with Mador de la Port.
Alle þis compayny of court com þe kyng nerre
For to counseyl þe kny3t, with care at her hert.
Þere watz much derue doel driuen in þe sale
Þat so worþé as > Wawan > schulde wende on þat ernde,
To dry3e a delful dynt, and dele no more
wyth bronde.
Þe kny3t mad ay god chere,
And sayde, ‘Quat schuld I wonde?
Of destinés derf and dere
What may mon do bot fonde?’
Once you learn the unfamiliar letters þ & 3 and a couple of others not in this passage, and learn different pronounciation conventions, it’s easier than you’d think to read & understand middle english, once you know the trick: The trick is to read it aloud and listen to yourself as you do so; that’s how the contemporary scribe expected people to read. Do that, and baffling passages like After þe sesoun of somer wyth þe soft wyndez suddenly turn into After the season of summer with its soft winds. The only tricky part is the word “þe”, (pronounced ‘thay’ with a hard th sound, like thomeone with a lithp ith thpeaking the word “say”), and that’s only because it’s a multi-purpose preposition that has several different meanings, some of which have different modern forms.
To summarise the definitions so far: ‘air’ (or ayr/ayre/aire etc) simply means ‘tune’ or ‘melody’. ‘Slow’ means ‘slow’.
So a ‘slow air’ is any tune played slowly.
The more interesting question is - which tunes sound good when played slowly?
If something is written that way - that is, the composer intends it to be a slow air - then it probably sounds good like that.
Many tunes have a ‘best tempo’, and don’t sound right at any other speed.
The majority of tunes, however, can be successfully adapted to a completely different tempo, and you can make some surprising discoveries by trying things at a very slow speed.
‘King of the Fairies’ is a lively hornpipe, but Dave Swarbrick makes a stunning slow air out of it. Other hornpipes will take the same treatment, but not all make equally good slow airs.
‘Spel-Gulle’s Polska’ is a Swedish tune my wife & I like to play really slowly, but actually we’ve no idea what speed it traditionally goes at.
If you know the old music-hall song ‘When Father papered the parlour’, you’ll know it has a brisk tempo. It also makes a haunting slow air.
Any other examples?
Other examples ?
As a song “Johnny stays long at the Fair” (aka “Oh dear what can the matter be”) is normally at a fairly steady 6/8, but becomes a waltz (? slow march) as “The Mist Covered Mountains” and on Mark Knopflers sound track to Local Hero was more of a slow air. Or back to 6/8 as adapted to be an irish jig.
I play the verses slowly and pick up tempo for the chorus for that one. Music-hall songs are great. My favourite is “She was only the stable-master’s daughter, but all the horsemen knew her”
In spanish, the exact translation of the english word ‘‘air’’ is ‘‘aire’’, so yes, that word exist, but in another language.
here’s a question: is there such thing as a “fast air”?
Not as a jargon term in use in ITM or any other trad circles or musicologically that I have ever encountered. It would sound a little odd to use the phrase as we rarely use the word “air” in its basic musical sense any more in modern English, but there would be nothing “wrong” in describing, say, a reel-tune as “a fast air for dancing a reel to” or some such. Since “air” is in musical terms just another word for “tune”, “melody” or perhaps “theme”, there’s no reason other than linguistic, subjective comfort why it could not be so used. As has been pointed out, the cognate word is still in current use in French and Spanish, (written “aire” but pronounced differently, of course, in each) neither of which languages has, so far as I know, a direct translation of the English “tune” either as a cognate word or with quite the same conceptual import. (I haven’t researched this, so please correct me/shoot me down!)
No, that misses the critical element mentioned by Nano and Nico and s1m0n. Namely, sean nós.
As Jem said, tunes are plastic, and they morph fairly readily from one dance form to another. Slow reels, slow hornpipes, slow jigs are recognized as such. Play the melody of a song, and you’re playing an air, at any speed. But an air played slowly is not necessarily a Slow Air.
In ITM parlance (which is what I think the OP here is referring to), the term “Slow Airs” or “Foinn Mhalla” usually refers to a style of performance that is informed by the traditions of sean nós singing. It’s not only the speed, but the phrasing, free rhythm, unique ornamentation, and evocation of sean nós vocal expression that makes them what they are.
As a player of dance tunes, I find that what I know does not prepare me to to tackle the playing of slow airs with any justice. But I’ll gladly whistle the melody of “Stairway to Heaven” or “The Mother’s Lament” for you, as slow as you like. “Oh, yer baby has gone down the plug hole …”
Maybe mouth music - Irish lilting or Scottish puirt a beul. Reproduce this on an instrument, and you’re playing a fast air, in effect. Except with the weird circularity of an instrumental performance of a vocal performance that is intended to imitate an instrumental performance in the first place. 
That seems more or less right, Jem. The OED cites a French definition of “air” from around 1600 as: suite de tons et des notes qui composent un chant. Larousse Unabridged still gives “air” as a translation of “tune” in a generic sense. English and French are presumably derived from Italian, where aria has both the generic and operatic meaning. In Spanish, “son” can also be a certain kind of tune in a Caribbean context.
Folk etymology would probably derive “air” from the breath of singing of blowing an instrument. But the OED speculates that the connection goes back to two senses of Latin “modus” - manner or mood, and mode in the sense of a musical scale. And that when aria took on the first meaning in Italian, it took on both and spread from there.
Funny that I should read this now, as just yesterday I played Chi Mi Na Morbheanna/The Mist-Covered Mountains on the Highland Pipes at a funeral.
The family was huddled under an awning barely large enough as there was a heavy steady driving mist- the ideal weather, in fact, in which to bury a grand old lady who had spent the first half of her 80-odd years in Scotland, up on a hillside or slope which on a clear day gives a magnificent view of the Pacific Ocean, and from which Catalina Island on such a day looms off the coast.
But on this day a heavy mist drizzled from the Heavens and the surrounding hills were indeed covered in mist.
I took it all in, and the proper tune to play could be nothing else.
But then I continued to play, in part because my pipes were going so well (they love this sort of weather!) and in part because the setting was so lovely. Flower of Scotland. Amazing Grace. Fields of Athenry (the family are Celtic supporters from Glasgow). Purple Heather.
Not till I got home did I realise how soaked I was! A bundle of wet smelly wool. Took quite a while to iron all the wrinkles out of the kilt’s 30-odd pleats.
how long did it take to iron the bagpipes?