Simply put, a Mook that flies, floats, or hovers, due to either having wings, a propulsion system, or supernatural powers. They may stay out of the player's reach, and will attack him from their advantageous position. Furthermore, due to their freedom in the air, they may tend to dodge rather well.
Because of this, they can be considered Goddamned Bats or, in worse cases, Demonic Spiders, especially in games where your character has Denial of Diagonal Attack or takes Knockback (see Ledge Bats for this case). Can be a type of Kung-Fu Proof Mook if they're especially hard to hit normally.
These guys usually tend to do one of the following:
Stay out of the player's reach and shoot them or drop bombs (or Mooks) from above.
Airborne Mooks in Platform Games, usually those that try to knock the player over, are considered a huge annoyance, especially if they're the kind whose sole purpose is to knock the player into a Bottomless Pit.
Airborne Mooks in FPS games or Action games may be smart enough to strafe around the player, making them harder to hit. Other than that, they use the same strategies above.
RPG games will tend to have these guys as Fragile Speedsters. They'll usually have a high speed and evade rate, making them a pain in the ass to hit. Thankfully, they probably won't have high Hit Points, and will go down quickly if you do hit them. In the case where they do have high Hit Pointsand a high evade rate (and maybe some very damaging attacks), you're fighting a Demonic Spider.
If these things appear in a Tower Defence game (especially those where you have to divert the mooks' path), they'll usually have the ability to take a short cut and fly over your towers to their goal. Certain types of towers will not work on them either.
Your best bet against an Airborne Mook is, obviously, to use an Anti-Air attack against them, as they'll usually be weak to it or are unable to avoid it.
Garden Gnome Carnage has sleighs. They float, and some of them drop gift-clad parachuting elves on you, but on the other hand, they can be helpful as they explode like a brick when they hit the ground, likely taking out some elves in the process.
In the Garden Gnome Carnage spin-off Hyper Princess Pitch, they return. They shoot projectiles at you, and some of them only fly by, leaving you only short time to kill them, while others stay around and you have to destroy them.
The Harpies from Serious Sam, and the Flying Kleers, Floaters, Hellchicks, and Levitators from Serious Sam 2. They all possess projectile attacks, whether they be fireballs (Flying Kleers), energy projectiles (Harpies, Floaters, Levitators) or bats (Hellchicks).
Halo has jump/jet pack Elites and Brutes, as well as Sentinels and Drones.
Action RP Gs
Vespoids in the Monster Hunter series. They are essentially giant wasps that hover just out of reach of many of the game's weapons, before darting in, jabbing you with a stinger, and retreating. Couple this with the fact that the stinger can inflict paralysis on you, and will seemingly always do this when you're low on health/fighting a boss level enemy, and you will soon come to hate them.
Also, they nearly always shatter when you DO kill them, leaving absolutely nothing to loot, making the whole exercise of killing them completely pointless (fortunately, Poison works fine if you need a body to carve).
The fact that the developers have included quests based entirely on slaying large numbers of these things (an early quest has you slay 20. At this point in the game, it takes 3-5 hits to kill one of the things!) also accentuates the utter irritation they bring. However, at least they spawn almost infinitely in certain places, so you don't have to go looking everywhere for them.
The Giant Mosquitoes in the Blighttown swamp in Dark Souls. Hard to hit, annoying things. They're easily killed by the even the weakest of attacks but here is the kicker; They endlessly respawn.
The Super Mario Bros.. games have: Lakitu, who flies out of normal range and drops spinies onto the player; Koopa Paratroopas, flying versions of the regular Mooks; and Bullet Bills, which try to ram the player, as some of the notable ones.
The Mega Man series had many Mecha-Mooks that could fly. Some of the more irritating ones were "Pipi", a robotic bird carrying an egg which it drops. If the egg hits the ground, it would break into 8 or so mini-birds, which would then fly at Mega Man. Especially annoying were those "things" that lived in Bottomless Pits which popped out from them as you jumped over, knocking you backwards and into the pit.
Metroid Prime features flying space pirates. And, of course, wasps!
Bug! also had at least one in each level, and they were usually very annoying to defeat.
Terraria has a number of them, including demonic eyes and eaters of souls.
Commander Keen 4 has Skypests. Can't shoot them, can only crush them with your pogo stick when they land.
Ninja Senki features one in every level group. The ghosts in sections 1 & 2, Purple Flames (these fly high and drop small flames onto the ground) at sections 3 & 4, demon heads going in circles around you on sections 5 & 6, etc., etc.
Castlevania gives us the gorgon heads. They don't usually turn you to stone, but they do fly in a sine wave when you're on platforms. If the platforms are moving, they becomes Demonic Spiders rather quickly.
A lot of the flying demon enemies in the Metroidvania entries also do this. The sheer amount of mid-air enemies makes the Axe subweapon very good in these games.
Snailiad has, among others, the Sky Viper, Chirpy, Batty Bat, and Ghost Dandelion.
Goddamned Zubats from Pokémon, or any Flying-type for that matter, although they weren't as annoying as the bats...
Pokémon with the Levitate ability to dodge Ground-type attacks can count too, though Koffing/Weezing are probably the most annoying, having a high Defense, being able to poison you or blow up, and having only one weakness because Levitate removes its Ground weakness...
The Tynamo family is particularly nasty for this reason; its offensive stats are high enough that it can do more than stall, and its weakness to Ground attacks is annulled by its Levitate ability, effectively leaving it with none. That is, unless Gastro Acid, Entrainment, Gravity, Mummy, or Mold Breaker come out to play.
Final Fantasy XII had some enemies with the flying status, which meant that you couldn't use melee attacks against them unless they were guns or bows/crossbows. Alternatively, one could just use magic on them without having to open up the inventory all the time.
Final Fantasy VII also had enemies with the "flying" status which could not be damaged by melee attacks. There were also bat enemies which had a 1/8 (later ones had 1/4) chance of completely avoiding a physical attack, making them literal Goddamned Bats.
The Mario & Luigi series has them. Hammers and other low-hitting moves will not be able to hit these kinds of enemies, use your Goomba Stomp instead. Bowser from Bowser's Inside Story has it pretty rough against the (very few) airborne enemies he fights - he is unable to attack them via any of his normal attacks.
Mario & Luigi: Dream Team adds mooks (and bosses) that change from being airborne mooks to land based ones mid battle (and vice versa), with Flibbees going between being right way up and on the floor and upside down and in mid air. There's also Robo Drilldigger, a giant boss who has both land and sky forms (and whom is immune to the jump or hammer depending on which one is in use) and the boss Pi'illodium, who can thankfully be taken out the air and made vulnerable to ground based attacks if you destroy its wings.
Flying Assault Drones and Rocket Drones in Mass Effect. This also made them immune to most biotic attacks. They land eventually, though.
A better example is the Geth Hoppers, although they don't fly, they're more mobile and spend more time above ground than the turrets.
Two kinds of flying enemies appear in Defense Grid The Awakening, one of which just has more health than the other. In any case, both of them are very dangerous as they are immune to some towers, and cores stolen by them cannot be replaced.