We like our wizards to fit a certain mold. They seem to have the same characteristics in many of our classic adventure tales. A wizard trains or mentors the young protagonist (to whom he is not related), has knowledge unavailable to the general public which was passed down from long ago, uses supernatural powers, sports a white beard (mainly to show his advanced age), and despite his age, he's still good in a fight.
Merlin the Magician is the oldest of the archetype. The character dates back centuries, from the writings of Geoffrey of Monmouth. The version we know best was taken from T.H. White's book The Once and Future King, which was the source material for Camelot and The Sword in the Stone. The other wizards follow in Merlin's footsteps.
Mentors: Arthur, who became King of Britain.
Knowledge: More educated than anyone around him.
Supernatural powers: Engineered Arthur's conception by sorcery, can see remotely, prophesied the outcome of various battles and Arthur's actions.
White beard: Yes.
Fights: Usually by magic only, although in some movie versions he can swing a sword with the best of them.
Obi-Wan Kenobi (Sir Alec Guinness) was a Jedi Master who became a hermit on the planet Tattooine in order to hide from the Empire and keep an eye on Luke Skywalker. He trained the young hero in the use of The Force when they were brought together by the droid R2D2. His role in training Luke was passed to the much older Yoda upon Obi-Wan's death.
Mentors: Anakin and Luke Skywalker. He was more successful the second time around, which goes to show that older wizards are better.
Knowledge: Jedi training.
Supernatural powers: Skillful use of The Force ("These are not the droids you're looking for.")
White beard: Yes, in episode four.
Fights: He was a master with the lightsaber, but could not bring himself to kill Darth Vader.
Gandalf the Gray (Sir Ian McKellen) made it his life's mission to rid Middle Earth of the evil influence of Sauron and the Rings of Power. Along the way, he advised various entities and brought them together to reclaim their lands.
Mentors: Frodo, Aragorn, and their comrades.
Knowledge: The entire history of Middle Earth, plus a limited ability to see the future.
Supernatural powers: Plenty, including returning from the dead.
White beard: Yes, although it was gray early in the saga.
Fights: Fearlessly, on horseback or on foot.
Albus Dumbledore (Michael Gambon) is the headmaster of Hogwarts School of Witchcraft and Wizardry.
Mentors: Harry Potter and other students at Hogwarts School.
Knowledge: Was educated at Hogwarts himself.
Supernatural powers: Producing fire, invisibility, various spells.
White beard: Yes.
Fights: He spent years battling Grindelwald and Voldemort by various magical means, most often with conjured fire.
Mr. Miyagi (Pat Morita) trained The Karate Kid by getting his household chores done ("Wax on; wax off"). He's a master at what he knows, and bluffs his way convincingly through anything he doesn't know.
Mentors: Daniel Larusso.
Knowledge: Ancient martial arts techniques, zen philosophy, and a trained poker face.
Supernatural powers: Only a magical healing touch, which was enough to save the day.
White beard: Just the minimum requirement.
Fights: Whipped five teenage black belts in 30 seconds.
You don't have to be human to be a wizard. Rafiki the monkey shaman kept an eye on the pride in The Lion King.
Mentors: He wasn't a long-term teacher, but he set Simba straight on how to live his life. Afterward, he became an advisor to the lion king.
Knowledge: He is the philosopher of the animal kingdom.
Supernatural powers: Could see remotely by gazing at feathers.
White beard: Yes.
Fights: The hyenas with his staff.
The word wizard isn't normally applied to Santa, but he fits the profile, with the exception of mentoring the protagonist. No one upstages Santa Claus!
Mentors: Nobody in particular, but he won't bring you toys unless you are a good girl or boy. That's a powerful incentive to behave.
Knowledge: He knows when you are sleeping. He knows when you're awake.
Supernatural powers: Drives flying reindeer, packs amazing amounts of gifts, rises up chimneys, and eats tons of cookies.
White beard: In abundance.
Fights: He prefers not to, but was arrested for assault in Miracle on 34th Street, and saved the day in Santa Claus Conquers the Martians.
Curiously, the character who actually uses the title Wizard (in The Wizard of Oz) doesn't display any of these characteristics! There are no doubt other characters who do fit this same profile. Can you add more to the list?