I've read that Mithrandir is one of Gandalf's aliases. Why do certain people call him by this name and not Gandalf? I've noticed in particular that Galadriel uses the name Mithrandir quite often.
Gandalf is one of the Istari - practically immortal creatures put in middle earth to guide and help its inhabitants. He is therefore very long lived and has picked up a lot of different names or nick-names to different people at different times.
Mithrandir is a Sindarin phrase meaning Grey Pilgrim or Wanderer (ref: Tolkien Gateway). Sindarin is the everyday language of Elves and therefore the name that Galadriel uses for him.
Gandalf, or Gandalf the Grey, or Gandalf Greyhame are just some other names for him - I don't think we are told that any of them is distinctly his name. Gandalf appears to be the common name used by Dwarves and Hobbits for him at the time of The Hobbit, hence the name we think of him as having.