A wraith is someone who has been killed yet lives on by some dark sorcery (necromancy or a Morgul-blade), or someone who died in darkness or serious dishonour (the latter are comparable to popular culture's notion of ghosts).
During Sauron's influence and rule in Middle-earth, he swayed and corrupted nine powerful Men of Númenórean and Easterling stock by giving them each one of the nine Rings of Power. Afterwards, they slowly faded into darkness, becoming Ringwraiths enslaved to the will of Sauron, fated to roam the world until their end.
No other evidence of any other Wraiths are recorded as existing in Middle-earth, save for the Ringwraiths, who continued in the East and in Mordor and when the Ring was discovered came out to search for it. When Frodo is stabbed by the Witch-King of Angmar in the first book he is infected by the evil magic from a Morgul blade; if Glorfindel had not saved Frodo, he would have eventually become a Wraith himself.