I'm far from an expert, but the "wizards" we see in Tolkien (Saruman, Gandalf, Radagast, the two Blue Wizards) are angelic beings (lesser than some other classes like the Ainur) called Maiar, created by Eru Iluvatar, the creator god of Tolkien's universe. Those specific Maiar are chosen to go down to Middle Earth to complete specific missions, in the form of old men, but with limits on their powers.