There is a great controversy over the name of Celestia and Luna's race. Winged unicorns aren't among established mythological creatures and there is no fixed name, but, obviously, we need to call them somehow. Below are several possibilities for people not satisfied with plain, descriptive "winged unicorn".
Excuse me for overusing "some people do X", but considering the nature of this work ("heard on the Web"), it may be very hard to pin-point sources.
It may be a little counter-intuitive, but the creators of Friendship is Magic went in this direction. After all, Luna is "the younger unicorn [who] refused to lower the moon to make way for the dawn." Similar decision was made by Madeleine L'Engle in A Swiftly Tilting Planet (1978).
"Alicorn" seems to be most widespread in My Little Pony fan community (mistakingly attributed to Dungeons and Dragons — see: Unisus), although meets fierce resistance from selected few. The main problem is that it already has an established meaning — unicorn's horn or the substance it is made of. The word may have possibly come from Italian "alicorno", which is usually translated as "unicorn".
There have been attempts to make up a false etymology for this word and derive it from Latin "ala" (wing). Standard Latin prefixes "ali-" and "ala-" are used to signify connection with wings or winglike shape. Therefore "alicorn" and "alacorn" could be understood as "wing-horn". "Alacorn" lacks the cultural significance of "alicorn" and some people choose it as preferred name.
Usage of "alicorn"
Mythology of the Spanish region Cantabria provides us with an example of a unicorn with winged hooves, called "El Alicornio". Folk etymology affirms that it is a portmanteau of "alitas" (little wings) and "unicornio", but it might just as well come from the aforementioned Italian word.
The first known usage of this word with our meaning (a hybrid of a unicorn and a pegasus) is in Piers Anthony's book Bearing an Hourglass (published in 1985). When asked how he came up with this term, the author gave a prosaic answer:
- "I saw fantasy statuettes in an ad, and one was of a winged unicorn, titled an alicorn, so I figured that was the name and used it. I have not been able to verify it elsewhere."
It seems that Piers Anthony's portrayal didn't stick among Fantasy writers. Andre Norton and Mercedes Lackey in their Halfblood Chronicles (first book, Elvenbane, published in 1993) portray alicorns as very ferocious unicorns, serving as elven war steeds.
In the same year the Dragon Magazine, an official magazine for source material for Dungeons and Dragons, published an article called "Unique Unicorns" providing additional creatures to use in adventures, alicorn among them:
- "An alicorn is exactly like a sylvan unicorn, but with a gnarled and twisted horn."
In Final Fantasy XI Auroral Alicorn is one of the monsters in the Monoceros family. They don't resemble unicorns, but each has a single curved horn on its forehead.
Compositions of Greek "πτερον" (pteron — wing) and κέρας (keras — horn). Some people try to combine Greek and Latin words into something like "pterocorn", but such practices are generally considered inappropriate.
These two are the most naive portmanteaus of "pegasus" and "unicorn", but have a merit of being easily recognized. One classification I've found is that pegacorn is a creature, whose father is a pegasus and mother a unicorn. In unipeg's case, the opposite would be true. Be warned, however, that this is most definitely not the way Celestia and Luna were born.
Unisus, another portmanteau, is the actual word 2nd edition of Advanced Dungeons and Dragons uses to name winged unicorns. The article in the Dragon Magazine gives us this:
- "Unisi are unicorns with pegasi wings, the result of crossbreeding between the two species. A single such creature is called a unisus."
Ethiopian pegasus (lolwut)
Perhaps the oldest name of all, even though it makes little sense, considering contemporary usage of "pegasus". This was the name Pliny the Elder used in his Natural History (published AD 77-79):
- "Æthiopia produces [...] many monstrous kinds of a similar nature; horses with wings, and armed with horns, which are called pegasi;"
African creatures were portrayed in medieval bestiaries, sometimes with one horn, sometimes with two. From more recent creations, a winged unicorn in Sailor Moon Super S was called Pegasus, even though his kanji name could be more accurately rendered as "Unipegasus".
- Shepard, Odell. Lore of the Unicorn (1930) 
- Licorne on French Wikipedia
- Word Information
- Mitología en Cantabria
- Wybo, Michael John II. "Unique Unicorns", Dragon Magazine (February 1993)
- Auroral Alicorn on FFXIclopedia
- Pliny the Elder, Natural History, book 8, chapter 30 
- List of minor Sailor Moon characters on Wikipedia