Nazgûl

The Nazgûl were the Nine Kings of Men corrupted by Rings of Power and fell in service to the Dark Lord Sauron.

Encounter in the Forest
The Nine Nazgûl had been sent from Mordor by Sauron to find Frodo Baggins. One of them caught up with the hobbits on the road and started searching for them, sensing the One Ring. When he found nothing, the Black Rider returned to his horse and left.

Bree-Town
Five of the Nazgûl learned of the hobbits' location and went to the town of Bree, entering the inn of the Prancing Pony. They mistook several pillows for their sleeping prey and proceeded to cut them to pieces until they realized their own folly.

Ambush at Weathertop
The same five Nazgûl tracked the hobbits to Weathertop where the Witch-king himself stabbed Frodo with a Morgul-blade, before they were driven off by Aragorn with a flaming torch.

On the road
All of the Nine regrouped and followed the group on their journey to Rivendell. The Nazgûl managed to force the wounded Frodo Baggins to break off from his companions and moved towards the river. When the hobbit refused to give up the Ring willingly, the Ringwraiths attempted to cross over, however a magical flood unleashed by the wizard Gandalf the Grey and Elrond stopped them from reaching their quarry.

Trivia

 * In Ralph Bakshi's The Lord of the Rings, the Nine Nazgûl reveal armor under their robes, unlike the J.R.R. Tolkien's novels in which they resemble ghosts when seen with the One Ring on.
 * They are never seen again after the flood, even of there was a scene in the book where the Fellowship of the Ring encounters a Nazgûl mounted on a Fellbeast while on the river. The creature is shot by Legolas and probably dies of it's wound, the Ringwraith being forced to flee.