Slay means to kill in a violent way. It has a mythic quality. You might slay the red knight. You might slay the dragon. You might slay the demon. But you probably wouldn't slay an annoying mosquito.
Slay is an old word associated with knights and dragons. Like many old words it has some funky forms. When you killed someone or something in the past you say slew, as in "St. George slew the dragon." When you've already done it, you say you have slain.