Their tactic for finding people with the ring is admirable. They programmed their robes with magic to disappear to anyone wearing the ring, and since they can see through the ring's invisibility power, all they have to do is A) look for people who think that they're invisible or B) look for people who stare at them in horror.
The Ringwraiths hunt down anyone who tries to stand in the way of Sauron, namely Frodo, The Fellowship of the Ring, Harry Potter and Darth Vader. Reports on the progess of the capture of Darth Vader are disappointing.
There are Nine Ringwraiths, with the Witch-King of Angmar being their leader. They often ride around on black horses, however, they switch these for giant flying fell beasts (also known as Nazgul, oddly).