Feywild
The Feywild, also known as Faerie, is a plane that exists as an "echo" of the Prime Material Plane. It is from here that the fey originate, giving the Feywild its name, and it is also a place of unusually potent magic.
It's pretty much the Points of Light rendition of marginal "Faerie-world" planes that have been mentioned (and then mostly ignored) in Planescape and Forgotten Realms. As such, it gets slightly more exposure in the 4e FR sourcebooks, since elves come from the Feywild and elves have a big role in the FR setting.
Traits
As a parallel plane to the Prime, the Feywild's geography is similar, though not entirely identical, to the geography of the Prime. Similarly, just as the geography is reminiscent of the Prime, so are its inhabitants and many creatures exist as fey "echoes" of the Prime creatures.
Arcane magic runs more freely and powerfully in the Feywild than it does in the Prime and it is for this reason that so many of its inhabitants are magically gifted. The Feywild is also reputably beautiful compared with either the Shadowfell or the Prime. The planes sourcebooks for Points of Light describes this intensity as as a result of being made from the "too bright" substance of creation; everything is intenser here, to the point that ordinary people may be at risk of dying from the sheer beauty (or awfulness) of the plane if they're too weak-willed.
Territories
Though much of the Feywild is the same as the Prime, geologically there are differences, and more importantly, the lands answer to different powers than those on the Prime (at least by default assumption). Many of the elven or eladrin civilizations have their earliest roots here, and abandoned remnants of their culture litter the landscape. Unlike the elves, however, who have all but abandoned the Feywild, some eladrin remain behind in the Feywild, most notably the "noble" eladrin. These eladrin still maintain strong kingdoms and citadels, many of which may spend at least part of the time in the mortal world.
Feydark
The Feydark is the Underdark of Toril's echo in the Feywild, and like most parts of the Feywild it is more majestic and fantastic than it's natural counterpart. Inhabited by the fomorians, a Fey echo of the titans, the Feydark is a cavernous world of maze-like tunnels and portals linking to the Underdark, courtesy of the efforts of Torog.
The most powerful of the fomorian realms is Mag Tureah, which stands within a country-sized cavern lit by precious gems and stones. This kingdom is ruled by the fomorian king Bres.
Sildëyuir
This territory only exists in the Forgotten Realms version of the Feywild.
Sildëyuir was once a demi-plane created by the star elves which connected to their ancestral home in the Yuirwood. Driven to the Feywild by conflict with the local humans the star elves retreated further and further from the Prime, and by the time of the Spellplague most of the star elves lived within Sildëyuir. Since then the Yuirwood has become rife with blue fire, preventing further travel between Sildëyuir and the Yuirwood. In spite of this immigration to Sildëyuir has continued through the intermediary of Myth Drannor and the realm is now the home to many elves and eladrin.
Sildëyuir is located within the same part of the Feywild that the Yuirwood occupies on Toril in the Prime. There the land is an untamed forest laid beneath a twilight sky, a wilderness broken only briefly by towering glass citadels built by the star elves and other eladrin.
History - General
The Feywild, also known as the plane of Faerie, was created by the Primordials, beings of power comparable with the gods. Finding some things of the Prime too "bright" or too "dark," the Primordials split the Prime into three nearly identical worlds: the Plane of Shadow and the Feywild.
History - Forgotten Realms
For a time the Feywild was located in close proximity with the Prime and the inhabitants of both planes interacted regularly. It was during this time that the eladrin and other fey first immigrated to Abeir-Toril.
Later, possibly due to the Sundering that created Evermeet, the Feywild slipped out and away from the Prime. Contact between the two planes ceased except through intermediaries or powerful arcanists and the plane was largely forgotten. This changed again, however, with the Spellplague. The destruction wrought by Mystra's death yanked the Feywild back into proximity, opening up new crossroads to the plane. Evermeet vanished, slipping into the Feywild, and in reverse new communities of fey creatures appeared throughout parts of Toril.
Likewise, now that the barrier between these two planes is much thinner, the eladrin discovered that their close bonds with that realm allow them to slip into and out of the Feywild.