what type of character is gandalf

There are several ways literary scholars classify, or group, characters. Sometimes we talk about protagonists (main characters) and antagonists (opposing characters). Other times we talk about static characters—characters who don't change much in a story and often play a short role. Finally, we talk about changing or dynamic characters, which are individuals who shift and change, often playing a more significant role in a story.
Gandalf is not a protagonist in our story; the story does not focus on him. He's a supporting character who comes alongside Bilbo and his friends on their journey to the Lonely Mountain. He plays a very significant role in the plot by encouraging Bilbo to accompany the dwarves and to act as their jewel thief. Gandalf constantly shows that he believes in Bilbo, though Bilbo begins the story with minimal courage. One time when we see Gandalf's belief in Bilbo's strength is when he says:

"In fact I will go so far as to send you on this adventure. Very amusing for me, very good for you -- and profitable, too, very likely . . ." (ch. 1)

Gandalf chooses Bilbo to go on this adventure to the Lonely Mountain. Even when the dwarves look down upon Bilbo, saying, "He looks more like a grocer than a burglar!" (ch. 1), Gandalf defends him.

"You asked me to find the fourteenth man in your expedition, and I chose Mr. Baggins. Just let any one say I chose the wrong man or the wrong house, and you can stop at thirteen and have all the bad luck you like, or go back to digging coal" (ch. 1).

Gandalf is continually supportive of Bilbo, reminding the dwarves that he has more bravery and strength than anyone initially thinks.
Gandalf accompanies them for portions of their journey, and he rescues them from their enemies at several points. For example, he protects Bilbo and the dwarves from the trolls, Bert, Tom, and Bill, who are trying to eat them in chapter 2.
Overall, Gandalf is a supporting character. He's important but not a protagonist. Gandalf is consistently good, selfless, and helpful to those around him. He does not change significantly in the novel, so we would consider him a static character. Even so, he plays a very important roll in The Hobbit.

Gandalf is a good wizard, a kind, often humorous, but wise leader. He recruits Bilbo, an unassuming Hobbit, to join a group of Dwarves on a journey because he discerns that Bilbo has potential. Gandalf is a very powerful wizard who could solve almost all problems himself if he wanted, but he prefers to develop the talents of other creatures when he can. 
Gandalf has a marked tendency to flit here and there to take care of other responsibilities. This leaves Bilbo on his own to cope with dangerous situations without his help. While this may seem unreliable on Gandalf's part, it allows Bilbo to learn to think for himself and hone his leadership skills. 
Adventure always seems to emerge alongside of Gandalf. As the text says:

Gandalf! If you had heard only a quarter of what I have heard about him, and I have only heard a very little of all there is to hear, you would be prepared for any sort of remarkable tale. Tales and adventures sprouted up all over the place wherever he went, in the most extraordinary fashion.