They are speculated to be a species of sea slug, much like the better known sea angels who share a similar morphology and as such would appear to be their closest relatives. They seem to be solitary, as no one has ever observed more than one at the same location at the same time. Many different abyss angel body plans have been observed, with different amounts of limbs, lobes on their "wings", and different facial features, or lack thereof. There has been much debate over the reason for this inconsistency, some say it is an example of sexual dimorphism, or that it could be different stages in a complex life cycle, while others argue, much like with many other "cryptid" sightings, the contradictions between different sightings, especially those with no hard evidence, is proof of their entire existence being a hoax.
Above: Two possible abyss angel sightings captured by a aquatic ROVs studying the ecosytems around deep sea hydrothermal vents. Their distinctive bioluminescent "core" can be clearly seen.
Abyss angels have most frequently been sighted around whale falls, the sunken corpses of whales which create temporary oases at the seafloor. Many different creatures have been observed feeding on whale falls, including crabs, octopi, and osedax (bone worms), and many deep sea predators come to hunt these scavengers. The role of abyss angels in this ecosystem is uncertain, however. Though clearly possessing what appears to be a digestive tract, as can sometimes be seen through their transparent bodies, they have never been observed feeding or being preyed upon. Some speculate abyss angels are simply timid creatures and most of their behavior only occurs off-camera.
Above: An abyss angel sighting near a whalefall (look carefully in the background). Here we see a different body plan, with two "legs" in addition to the tail, as well as differently shaped wings.
The most interesting currently avaible footage of an abyss angel would have to be this video, which was recorded during scientific research of a whalefall. The abyss angel can be seen swimming into and then out of frame, seemingly aware of the fact it has been noticed. This footage was apparently once broadcast as part of a TV documentary about obscure deep sea creatures, though no other footage of said documentary exists online.
This footage was provided by a friend of mine who is currently doing research into obscure lost films. By their request I will provide a link to their page when that's all set up!