In Middle-earth, Elves were born as Elves, Hobbits as Hobbits, and so on. Were the Wizards born as Wizards, or were they young men that somehow evolved as they matured into Wizards?
Like Raphael says, Tolkien's wizards, called "Istari" by the Elves, were not men but lesser deities (Maiar) of the same kind as the Balrogs and Sauron himself. That means that yes, Gandalf is essentially the same kind of "creature" as Sauron or the Balrog from Moria!
The Istari weren't always in Middle Earth, but were instead sent by the Valar to help Men (presumably against such perils as Sauron and other servants of Morgoth). Gandalf succeeded in his task; Saruman failed.
I don't know if there are traditional "wizards" in Tolkien's world, but I suspect there can't be. Magic in Middle Earth is hinted to be not really magic, but the "natural" properties of each type of being. Not something you can study and become proficient at...
In Appendix B of LOTR Tolkien writes during the Third Age that:
When maybe a thousand years had passed, and the first shadow had fallen on Greenwood the Great, the Istari or Wizards appeared in Middle-Earth.
They were sent to contest the power of Sauron and unite Men and Elves but
...they were forbidden to match his power with power, or to seek to dominate Elves or Men by force and fear. They came therefore in the shapes of Men, though they were never young and aged only slowly, and they had many powers of mind and hand.
This would be circa 1100 according to the timeline given in the appendix.
As others have noted the Istari are similar beings to Sauron.
It's quite incorrect to say that the Wizards were even born, because they weren't. The Ainulindale (in the published Silmarillion) gives their origin:
There was Eru, the One, who in Arda is called Iluvatar; and he made first the Ainur, the Holy Ones, that were the offspring of his thought, and they were with him before aught else was made.
Eru is the Supreme Deity of Tolkien's work (indeed, Tolkien frequently refers to him as "God" in his letters and other writings) and he made the Ainur, even before making the world.
From there the Ainur participated in the creation of the world, and some of them went down into it after it's creation. Of this group the more powerful were known as the Valar, or Powers of the World (from LotR you may recognise this word from Damrod's cry during the Mumakil attack in TT: "Ware! Ware! May the Valar turn him aside! Mumak! Mumak!"); the lesser Ainur were known as the Maiar, or Beautiful Ones.
Each of the Valar had a number of Maiar in his or her retinue, and it was from the ranks of these Maiar that the Wizards came.
- Manwe, King of the Valar, supplied Gandalf, who also has association with Irmo (master of visions and dreams) and Nienna (mistress of pity and hope).
- Aule, the Smith, supplied Saruman. It's of interest that Sauron was also originally from the Maiar of Aule, who really needs to be spoken to about disciplining his followers.
- Yavanna, the Giver of Fruits, supplied Radagast, and text in Unfinished Tales suggests that he was pressed on Saruman as a companion, which may help explain Saruman's dislike of him.
- Orome, the Huntsman, supplied the two Blue Wizards.
So the Wizards were actually beings that existed before the world and came into it after it's creation, and were not born but made.
The wizards were originally Maia, some kind of lesser gods. I do not know wether anything is said about their origin; as far as Middle Earth (or rather Eä) is concerned, they have always existed.
As a previous answer said, the Istari were not born. They were created before Ëa and Arda, and were created as disembodied spirits, as far as I understand the text.
The Valaquenta in The Silmarillion mentions Gandalf's origins, albeit rather briefly. His real name is Olòrin. Before he became a wizard, he was a Maia who was especially fond of the Vala Nienna, "she who weeps", who teaches strength in sorrow; she grieves for the hurts of the world, and those who listen to her learn pity (one of Gandalf's defining characteristics). After the Elves awoke, and before Men awoke, Olòrin walked among the Elves, whom he loved, but always did so without physical form, or in the form of an Elf.
Much later, after Melkor was defeated, and Sauron rose to take his place, Olòrin was still in the West. He was ordered by the Valar to go to Middle-earth and help the Children of Ilùvatar fight Sauron. He didn't want to go, because he was afraid of Sauron, but the Valar (specifically Manwë, I believe) said that was all the more reason to go. He reluctantly accepted the decision, and went.
Only when the specific Maiar chosen for this task entered Middle-earth did they really become wizards, or Istari. They were not born, having existed since before the creation of the earth. They simply took on physical form, appearing as old men.
I've only just begun to read The Silmarillion, so I don't have all the answers, but I will revise this answer as I get more information.
I'm fairly sure I recall that the Istari arrived at the Grey Havens in boats, already in the form of old men. The two Blue Wizards arrived together and went off wherever it was they went. Saruman and Radagast arrived together (to the disgust of the former!) although I do not recall whether this was before or after the Blue Wizards. Gandalf arrived last and alone, as I recall, and was greeted by Cirdan who gave him the Ring of Fire.