Short answers, pulled from the story.
A barrow-wight is a wraith-like undead creature in Tolkien's Middle-earth that haunts burial mounds and animates the bones buried inside them. Tolkien based the idea on Norse mythology, where heroes in various sagas battle undead beings called draugrs.
Barrow-wights live in the Barrow-downs, where the Witch-king of Angmar sent evil spirits to prevent the restoration of the ruined Dúnedain kingdom of Cardolan. These spirits animated old Dúnedain and Edain bones already buried in the mounds there.
The barrow-wight, along with the character Tom Bombadil, first appeared in Tolkien's poem The Adventures of Tom Bombadil, published in the Oxford Magazine on the 15th of February 1934.
Peter Jackson cut barrow-wights entirely from his film trilogy. Scholar Brian Rosebury argues the cut is acceptable since it does not fundamentally change the story, though John D. Rateliff notes it forces Aragorn to hand the hobbits their swords at Weathertop instead.
Frodo Baggins resists the wight's spell, cuts off its hand with a small sword, and calls for Tom Bombadil. Bombadil expels the creature from the barrow and recovers its hoard of ancient Númenórean swords.
VFX supervisor Jason Smith designed the barrow-wights for the show's second season, describing them as ancient, reanimated heroes acting for evil against their will. Barrow-wights have also appeared in games such as The Lord of the Rings Roleplaying Game and in Khraniteli, a 1991 Russian adaptation of The Fellowship of the Ring.