Eladrin

An Eladrin, drawn in the Tony DiTerlizzi goodness of Planescape.

Elves who live in either Arborea or the Feywild and the Material Plane, depending on the edition.

In Planescape and 3e

They are a sort of Elven Chaotic Good celestials with a fairy motif. They are the natives of Arborea: a wild and free people who celebrate life and the arts. They believe in freedom and one's own right to determine what to do with this. They oppose the fiends in their attempts to subdue the peoples of the Material Worlds, aiding in non-violent ways and inspiring great things in the people. They seldom stay in one place for long, traveling both on their own plane and to other worlds to experience their bounty. This doesn't mean that they are weak or cowardly: an Eladrin roused to anger is a fearsome sight to behold and will rip a berk in half before they know it. Alongside their powers they are all able to take on a secondary form, frequently as some kind of ball of light that grants them additional abilities.

They exist in several types, between whom there is no promotion as with other Outsiders such as Archons and Baatezu: they appear in one form and stick with it. There are seven known types of Eladrin, four Lesser and three Greater, here noted from the weakest to the most powerful.

Coure

Resembling 2' sprites with gossamer wings, the Coure are messengers and scouts who love to play pranks on people and cause mischief. If they are angered however these jokes can turn lethal, with the creatures being capable of deadly hit-and-run attacks when discorporating into a 6" ball of light to build a safe distance to attack from.

Noviere

Living on the second layer of Arborea, the Noviere are aquatic beings who navigate the waters of the layer with great ease. They like to explore the Planes and the Material Worlds to find new coasts. Their skins and hair are tints of green, blue or gold. They lack the webbing or gills that you would expect from an aquatic elf and they can move with great ease on land. They can be rather fickle and flighty but mean well, and are rather approachable despite being difficult to talk to. They can turn into a form of shimmering golden water in the shape of a dolphin, in which form they are tougher and can drown opponents in their bodies.

Bralani

The Bralani live on the third layer of Arborea, Mithardir. They are survivors living in the deserts of the plane, making their lives harsh but more worthy to be celebrated. Bralani suffer from powerful mood swings and can be dangerous to approach, but if met in a peaceful mood they are welcoming hosts. They can turn themselves in living whirlwinds to tear at their opponents with powerful gusts of wind and blasts of either sand or snow, depending on their surroundings.

Shiere

The knights of the Eladrin, they fight with strength, honor and skill. They are tall and lanky beings that might not appear powerful, but they are as strong as any mortal warrior. Shiere sometimes gather in groups to explore the plane and clear it of intruders. They can change into spheres of light, but they can do so only once per hour, so they use this ability sparingly.

Firre

The Firres (pronounce: feers) are devoted to the ways of beauty, art, music and magic. They have red hair and eyes that burn with their inner flames. Their music is of unmatched beauty, making them desirable guests amongst the courts of the Faerie Lords. Firres are prone to wanderlust and frequently visit the Material World and the other planes, often finding good pay in the city of Sigil. They can turn into great balls of fire, burning all that the Firre can strike.

Ghaele

These Eladrin are the most concerned with fighting the threat of the Fiends. They travel to mortal worlds and seek those whom they can guide to performing great deeds of heroism. They do not reveal themselves, even after the victory of good over evil, instead finding a new place to defend the Material Worlds. They can turn into many-colored spheres, searing the enemy with great beams of brilliant light.

Tulani

The Faerie Lords are powerful beings who move their courts amongst the surface of the first layer of Arborea. They are beings of inhuman grace and beauty, of great magical and martial power and have great respect amongst the other Eladrin. They are also rather haughty and are easily offended but if their guests behave they are gracious hosts. Tulani can conjure great weapons out of thin air and can copy any secondary form that the other Eladrin have at double power. They are incredible dangerous beings who tolerate no evil, and answer to nobody aside from the Morwen, the Queen of Stars and the gods living on Arborea.

In 4e

In 4e, they absorbed all of the old "elf as a super-magical being with mastery over arcane arts" lore and stuff, since it was always kind of weird that elves got portrayed as being both wizard masters and druidic-styled nature-hugging hippy freaks, since wizards and druids often don't get on. Not that the split was entirely unprecedented, 4e was just the first edition to fully split the two instead of making each race simply "lean more towards" one side of the equation. They had Intelligence bonus, the more "magical" elf abilities like trancing and charm resistance, and an encounter teleport power named Fey Step. They're no good at anything physical, so if you want a mage/druid/etc. they're pretty sweet. On the other hand, they're a kind of Elf.

Dragon Magazine #384 introduced a subrace of Eladrin in the form of the Winterkin. These eladrin, as their name suggests, descend from servitors of the Prince of Frost and other members of the faerie faction known as the Winter Court. Marked by paler coloration, including a pronounced tendency to albinism, Winterkin are distinguished from normal eladrin by taking the feat Winterkin Heritage; this grants them cold resistance and a new racial power in the form of Winter's Shroud, a reactive teleport to being attacked that grants them concealment afterwards. Feats further flesh out the Winterkin's unique powers; Chilling Presence lets them use Winter's Shroud to inflict cold damage instead of teleporting, Ice Walk lets them ignore ice & snow-based difficult terrain, Swirling Snow lets you create a small zone of obscurement when you use Winter's Shroud, Winter's Reach expands the teleporting distance of Winter's Shroud, and Winter's Heart means that enemies who hit you in melee are slowed for a turn.

The same issue also presented a new racial Paragon Path, the Bralani Wintersoul, which represents an eladrin - Winterkin or normal - who has served the Winter Court loyally enough to be granted magical powers over ice, snow and freezing winds.

The Neverwinter Campaign Setting also presented rules for tweaking the Eladrin to more accurately represent the Sun Elves and Moon Elves of traditional Forgotten Realms, if you felt you really needed to.

In 5e

They appear in the 5e DMG as a fourth elven subrace. Pretty effectively translated, actually; +1 Int, Elven Weapon Proficiency, even got Fey Step (Misty Step spell-like ability usable once per short rest) back. If this seems a little underwhelming, being literally just "High Elf, but with Misty Step once per short rest instead of 1 Wizard cantrip of your choice", that's because 4e Eladrin were basically just High Elves honestly focused on arcane magic.

An alternate take on them appeared in the Unearthed Arcana for September 2017; this version can choose between either a +1 Intelligence bonus or a +1 Charisma bonus, marking the first ever return of 4e's flexible ability modifiers. Assimilating a little of their original fey-angel lore, Eladrin are now presented as magically attuned to the four seasons, which resonates with their various personalities. Mechanically, this means that not only do they retain their Fey Step as a "teleport 30 feet to a spot you can see once per short rest" trait, but now, whenever they complete a short rest, they choose which season to align themselves to and gain a cantrip based on that season: Friends for Autumn, Chill Touch for Winter, Minor Illusion for Spring and Fire Bolt for Summer.

This actually isn't as unprecedented as you might think; back in 4e, the Eladrin had a subrace called the Winterkin, representing eladrin who were still bound to the Faerie Prince of Frost and so had a natural, mystical connection to frost and snow. This new UA version just expands upon that.

Unfortuantely, because they were a massive labor of love on the part of the designers, and they had to share an update with the entire gith race, whom the designers admit in the accompanying video (which literally contains more footage of Mike Mearls gushing about how much he loves the eladrin and wishes to play them than on both gith subraces combined), they are also more-or-less responsible for the uninspired, low-effort rendition of those, and get a bit of heat for it.

In Pathfinder

Here they go by the name of Azatas, varying from "elves with super magical powers" like Bralani and Ghaele to more exotic forms like Lillends (lamias with angel wings), Brijidines (flying elves made of lava) and Lyrakiens (standard butterfly-winged fairies).