Gargoyle
Gargoyles and their cousins grotesques were statues of ugly humanoid monsters, sculpted demons that were popular decorations for churches and, to a lesser extent, the homes of the rich and powerful. The idea was that with their sheer ugliness, they would frighten away actual evil spirits, so they were very popular. Gargoyles are considered an epitome of the Gothic style, and tend to show up all over the place in settings with a Gothic aspiration, especially in "gothick-punk" settings like the old World of Darkness or Gothic Fantasy settings like Ravenloft.
Technically speaking, a gargoyle is a specific kind of grotesque, having been sculpted around and incorporating a waterspout. Few people remember this who aren't history buffs.
Fantasy Gaming
Gargoyles are a common feature in fantasy games, where they are usually depicted more as grotesques. They are often creatures of living stone, similar to golems in many ways, but intelligent, often mimicking living creatures in traits like needing to feed and reproduce, and usually evil - ironically, given the actual gargoyles were supposed to be protectors. In Dungeons & Dragons and Pathfinder specifically, they are actually a breed of lesser Earth Elemental, originally hailing from the elemental planes before miograting to the easier, more prey-rich environment of the Material Plane.
In 5th edition D&D, the fact that gargoyles are earth elementals who can fly is explained as their being the spawn of Ogremoch, the Archomental of Evil Earth, who created them in mockery of the Air creatures he so hates.
Warhammer 40,000
A gargoyle is a winged variant of the termagant and possesses the same armament, except for a venom gland that allows it to spit acid. While being weak and cowardly, shying away from combat without a synapse link to nurture it, gargoyles are described in the fluff as being lethal in a confined space, with claws, spikes, tails and flapping wings and venom flying everywhere. They typically like to dive bomb exposed and weakened units like Guardsmen command squads, as they are not able to defend from a swarm of giant, acid spiting, gun toting bats that flock you like you're simply the most tasty piece of icing on a cake and flying off again. They love units like command squads, artillery commanders, heavy weapon teams and even snipers. Anything alone and unable to fight back at mid and close range a labelled "dumplings" to them.
Basically, imagine the Bio Raptors from Pitch Black. But they can see. And light doesn't hurt them. And they have guns. Yeah, pretty fething scary.
On the Tabletop
Termagants with wings and a hefty points increase. Rather unpopular, but not any more so than the rest of the Tyranids 5th ed. Fast Attack choices. You need to know what you're doing to get the mileage out of these guys.
In game they have one to two uses. One is to have them Deep Strike right behind enemy lines, away from fighting, and then attack things like command units, snipers, heavy weapon teams, and gun emplacements. They will wipe out these things through fleshborer fire (takes around 20 to start working), and if the unit is down to two models or one wound, dive in and use their acid to finish them off.
The second use is to take a unit of 30, then have it deep strike right next to your flying hive tyrant. Then your tyrant can have cover, as well as another 30 attacks to aid him. A smashing combo uses two winged hive tyrants with twin-linked devourers, and 30 gargoyles each, to rape the living shit out of enemy lines. Place them where the enemy has a crap load of immobile artillery and vehicles with weak-to-moderate rear armour and then pelt them to death with so many die rolls they drown in them, and it takes a whole new agency of the administratum to process the results. Works well against Tau and Imperial guard.
Really though, gargoyles are simply bats with guns, and use numbers (like all Tyranids) mixed with speed to force an opponent's death from aforementioned dice drowning. Dakka on wings. Gargoyle models also are now plastic, not the hellish metal ones. Gargoyles would be better if they could use devourers. But then flying dakka bats with acid spit would be absurd. Unless Matt Ward wrote it. Update: Dakka spit bats have now been done: See the Hive Crone for the shining example of loldom.