No, Gandalf is not an Elf. While Elves are the Firstborn, the first creatures to be created on Arda, the world of Middle Earth, Gandalf is one of the Maiar, beings somewhat comparable to angels, that were the servants of the Valar, the sort-of gods in the Middle Earth cosmology. The Valar created the world, while the Maiar were their helpers and assistants, and in some cases heralds, standard bearers, messengers or captains.
There is a reason they seem similar, though. The Maiar are spirits, not locked into a single mortal body. When Gandalf, who was originally known as Olórin, went into Middle Earth, he chose a form that is similar to that of the Elves, to fit among them.
Here is how he is described in the Valaquenta, the second part of the Silmarillion, wherein the Valar and Maiar are described:
Wisest of the Maiar was Olórin. He too dwelt in Lórien, but his ways took him often to the house of Nienna, and of her he learned pity and patience [...] for though he loved the Elves, he walked among them unseen, or in form as one of them.