I think Gandalf is basically an angel in human form. In Tolkein's world there were two orders of angelic beings; the Valar, who ruled Arda; and the Maiar, their servants and helpers. Gandalf, along with Saruman, Radagast, Alatar and Pallando (a.k.a. Wizards/Istari), were Maiar beings sent to Middle-Earth to help the faithful Men and Elves.
Dumbledore seems to be just some dude in a dressing-gown with a magic stick. He still seems powerful though.