Gandalf wasn't an elf, but rather he was one of a few wizards sent to help Middle Earth. So, how did he end up with one of the Elf Rings? And, why did Frodo not notice it, when he easily noticed Galadriel's ring?
Gandalf was given Narya by Círdan the Shipwright because he believed that Gandalf had the highest inner greatness of all the Istari (Wizards). The scene is described in The Silmarillion:
Take now this Ring, for thy labours and thy cares will be heavy, but in all it will support thee and defend thee from weariness. For this is the Ring of Fire, and herewith, maybe, thou shalt rekindle hearts to the valour of old in a world that grows chill.
There's more about this in Unfinished Tales, such as Saruman's knowledge of Gandalf's possession of the ring:
And the Grey Messenger [Gandalf] took the Ring, and kept it ever secret; yet the White Messenger [Saruman] (who was skilled to uncover all secrets) after a time became aware of this gift, and begrudged it, and it was the beginning of the hidden ill-will that he bore to the Grey, which afterwards became manifest.
Gandalf hid the fact that he had the ring, unlike Galadriel or Elrond, and so it was assumed that Círdan was the holder of the third Elven ring (until the ring bearers left Middle-earth). At the end of Return of the King, we are told:
As he turned and came towards them Frodo saw that Gandalf now wore openly upon his hand the Third Ring, Narya the Great, and the stone upon it was red as fire.
We aren't told exactly how Gandalf concealed the ring; presumably some sort of magic is involved.
The ring's bearer between its creation and Círdan passing it to Gandalf is not clear: Unfinished Tales says that Círdan received the ring from Gil-galad, who received it from Celebrimbor, but The Lord of the Rings says that Círdan received the ring directly from Celebrimbor, at the same time that Galadriel received her ring.