'Wand Elf' (but note that this name was given mistakenly by Men; Gandalf is a Maia, not an Elf)
ga'ndalf (A note in Appendix E of The Lord of the Rings that final 'f' should be pronounced 'v' leads some to mispronounce this name 'gandalv'. 'Gandalf' is not an Elvish name, however, and this rule does not apply)
In origin a Maia of Manwë and Varda, Gandalf came to the northwest of Middle-earth after a thousand years of the Third Age had passed, with four others of his order. At the Grey Havens, Círdan entrusted him with the Red Ring, Narya, to aid him in contesting the will of Sauron.
Gandalf wandered widely in Middle-earth, and learned much of its races and peoples. Unlike his fellow Wizards Saruman and Radagast, he never settled in a single place. He was instrumental in the victory of the War of the Ring, but during that conflict he battled with a Balrog, and though he was ultimately victorious, his spirit left his body, but was sent back to Middle-earth to complete his task.
Gandalf finally left Middle-earth in 3021 (Third Age), when he departed over the sea with the Ring-bearers.
The Wizards arrived in Middle-earth after the end of the first millennium of the Third Age. They were sent by the Valar to aid Elves and Men, but none knew this but Círdan the Shipwright, Lord of the Grey Havens where their ships put in. Though Saruman was at that time the acknowledged leader of the Wizards, Círdan saw that Gandalf was in fact the greater, and secretly gave him the Red Ring Narya to aid him in his quest.
For almost 1,500 years, Gandalf wandered the northern and western regions of Middle-earth, learning of its lands and cultures. He was closest in friendship with the Elves, and especially Elrond, who had learned of his origins from Círdan.
At almost the same time as Gandalf had arrived in Middle-earth, a dark power had appeared in southern Mirkwood, at the evil place known as Dol Guldur. After a thousand years of his wanderings, Gandalf went there and drove the darkness into the east for a while. Soon it returned, and in 2463 (Third Age), the greatest among Wizards and Elves formed a White Council, with Gandalf as a prominent member, to counter the growing threat.
In 2850, he revisited Dol Guldur, to find that the power that held it had grown indeed during the eight hundred years since his last visit. He recognised the dark power now as Sauron returned, and escaped to inform the White Council. Before he escaped, though, he found Thráin II broken in the pits of Dol Guldur, and though Thráin died before Gandalf could help him, he did surrender a map and a key into the Wizard's keeping.
The Hobbits first appeared in the records of Men and Elves shortly after the arrival of the Wizards, but of all the Wise, Gandalf was the only one to pay them great heed. After the foundation of the Shire, he would visit periodically and was responsible for '...quiet lads and lasses going off into the Blue for mad adventures' (The Hobbit, Chapter 1, An Unexpected Party). Given this description, we are forced to wonder whether he had a hand in the exploits of Hildifons and Isengar Took, both of whom are recorded as having become involved in adventures of this kind.
Of all the hobbit families, Gandalf seems to have been most closely associated with the Tooks; he was a close friend of Gerontius, the Old Took (Frodo Baggins' great-grandfather), and was said to have given him a gift of magical diamond studs, which fastened and unfastened on command. After Gerontius' death in 2920 (Third Age), he was not seen in the Shire for more than twenty years, until he returned with Thorin and the Dwarves to involve Bilbo Baggins in the Quest of Erebor.
Like all the Wizards, Gandalf had the appearance of an old man, who grew old slowly with the passage of the centuries. His hair was white, and his long white beard grew down below his waist. His eyebrows were particularly noticeable; they were so long and bushy that they stuck out from beneath the rim of his hat.
He dressed in a long grey cloak, which is perhaps the origin of his title 'The Grey'. He is also described as wearing a tall shady-brimmed pointed blue hat, a silver scarf, and long black boots. He carried at all times a spiked staff.
The Magic of Gandalf
Tolkien at no point defines what the limits of Gandalf's magic were. As a Maia, he had many natural abilities that would seem magical to mortal races, but he also had a great store of knowledge of more 'mechanical' magic, worked through spells and incantations, and especially through the agency of his staff. It is clear that he had far greater power, especially after his return as Gandalf the White, than he ever displayed in Middle-earth.
His magical powers seem to be particularly associated with fire, a fact that is perhaps related to the Ring of Fire, Narya, that he bore.
A Gandalf Miscellany
Gandalf adopted the habit of pipe-smoking from the Hobbits. He often used his magical arts while smoking; he could change the colour of his smoke-rings, or send them in any direction he pleased.
Gandalf's preferred drink was red wine.
This is based on a note given in Unfinished Tales associating each of the five Wizards with one or more Valar, which connects Gandalf's name explicitly with Manwë and Varda. There is good reason to see this as definitive (for example, the same list also connects Saruman with Aulë, who is well established as that Wizard's original 'patron' Vala) but it is not absolutely conclusive. There are hints in other sources that make Gandalf ultimately a Maia of the people of Lórien, otherwise known as Irmo. On balance, though, the association with Manwë and Varda seems most likely.