Barrow-Wights were evil spirits from Angmar, who infested the Great Barrows after TA 1636. The wights would try to trap people in their barrows and sacrifice them.
When one caught Merry, Pippin, Sam and Frodo was the first to awake, as he glanced to the side he saw his companions,
They were on their backs and their faces looked deathly pale; and they were clad in white. About them lay many treasures, of gold maybe, though in that light they looked cold and unlovely. On their heads were circlets, gold chains were about their waists, and on their fingers were many rings. Swords lay by their sides, and shields were at their feet. But across their necks lay one long naked sword.
(The Lord of the rings)
Frodo became afraid and eventually heard a chilling song.
Cold be hand and heart and bone,
and cold be sleep under stone:
never more to wake on stony bed,
never, till the Sun fails and the Moon is dead.
In the black wind the stars shall die,
and still on gold here let them lie,
till the dark lord lifts his hand
over dead sea and withered land.
Then as he looked around he saw,
...a long arm was groping, walking on its fingers towards Sam, who was lying nearest, and towards the hilt of the sword that lay upon him.
(The Lord of the Rings)
Fortunately for the hobbits Frodo rebreed a song that Tom Bombadil had taught them to sing if they got into trouble on his lands.
Ho! Tom Bombadil, Tom Bombadillo!
By water, wood and hill, by the reed and willow,
By fire, sun and moon, harken now and hear us!
Come, Tom Bombadil, for our need is near us!
And as if by magic, there appears Tom, and banishes the wight,
Get out, you old Wight! Vanish in the sunlight!
Shrivel like the cold mist, like the winds go wailing,
Out into the barren lands far beyond the mountains!
Come never here again! Leave your barrow empty!
Lost and forgotten be, darker than the darkness,
Where gates stand for ever shut, till the world is mended.
And wakens the hobbits,
Wake now my merry lads! Wake and hear me calling!
Warm now be heart and limb! The cold stone is fallen;
Dark door is standing wide; dead hand is broken.
Night under Night is flown, and the Gate is open!
Hey! now! Come hoy now! Whither do you wander?
Up, down, near or far, here, there or yonder?
Sharp-ears, Wise-nose, Swish-tail and Bumpkin,
White-socks my little lad, and old Fatty Lumpkin!
What a jolly old fellow. I don't think that there is ever a description of an actual Barrow Wight, just a dismembered arm. Though I imagine a skeletal figure clothed in a long white robe is fitting enough, Tolkien seems to have left it up to our own imaginations. There is no more mention of Barrow Wights in The Lord of the Rings.