Gandalf means "staff Elf". This name was given to him by Men in reference to the staff he carried and the mistaken belief that he was one of the Elves.
Gandalf the White:
When Gandalf was sent back to Middle-earth after his fight with the Balrog, he was clad in white.
Olorin:
Olorin was the name by which Gandalf was known as a Maia spirit in Aman.
Grey Messenger:
When Gandalf arrived in Middle-earth he was referred to as the Grey Messenger because he was a messenger of the Valar.
Mithrandir:
Mithrandir was the name given to Gandalf by the Elves. It means "Grey Wanderer" or "Grey Pilgrim" and refers to his color.
Tharkun:
Tharkun was the name by which Gandalf was known among the Dwarves.
Incanus:
Gandalf was called Incanus by the peoples of the South. The origin of this name is not certain.
Gandalf Greyhame:
Gandalf was called this by Eomer of Rohan.
The White Rider:
Gandalf was first called the White Rider by Aragorn in contrast to Sauron's Black Riders.
About Gandalf
Gandalf was one of the Order of Wizards sent to Middle-earth to oppose Sauron. During his 2,000 years in Middle-earth, Gandalf came to understand and love the land and its peoples, particularly Hobbits. Gandalf worked ceaselessly to counter Sauron's plans to dominate Middle-earth, and he set in motion a chain of events that ultimately resulted in the Dark Lord's downfall.
Gandalf was one of the Maiar, spirits who helped and served the Valar. When he lived in the Undying Lands he was known as Olorin, and he was the wisest of the Maiar. He sometimes walked among the Elves unseen or as one of them and shared wisdom and fair visions with them. For much of the time, Olorin dwelled in Lorien, the gardens of the Vala Irmo for which the woods of Lothlorien in Middle-earth were named. Olorin also frequently visited Nienna in her home in the far west near the walls of the world. It was said that Olorin learned pity and patience from Nienna. But Olorin seems to have been most closely associated with the Valar Manwe and Varda, and it was they who sent Olorin to Middle-earth around the year 1000 of the Third Age.
The Maiar who were sent by the Valar to Middle-earth were given the bodies of old men and were known as Istari, or Wizards. Their mission was to help the free peoples of Middle-earth oppose Sauron without seeking domination or power for themselves. Olorin was the last of the five Wizards to arrive at the Grey Havens in northwestern Middle-earth. There Cirdan the Shipwright gave him Narya, one of the Three Rings of the Elves.