Watching Samuel Fuller's Pickup on South Street the other day, I noted that Fuller was extremely fond of a type of camera movement that wasn't very common before he came along: the super-fast tracking shot, where the camera very quickly moves in and up on a character, going from long shot to close shot in seconds. This kind of fast camera movement became a specialty of Martin Scorsese, a Fuller admirer. It is, like Fuller's movies themselves, quick, punchy, unsubtle and exciting; instead of the decorous, elaborate camera moves favored by a Vincente Minnelli, Fuller just puts the camera on wheels and shoves it in his characters' faces. His camera moves the way his scripts sound, if that makes any sense.
Camera movement is often used just as just another piece of film grammar; many "classic" directors, especially those who had started in the '10s or '20s (when elaborate camera moves could be rather difficult, technically), preferred to move the camera only to call attention to something in a scene or to emphasize a character's reaction, e.g. character X sees something he doesn't like, and the camera moves in on him to show that this is a big deal for him. Ford, Hawks, Lubitsch, Renoir and DeMille are just a few of the famous directors who either don't move the camera much or move it only for fairly obvious reasons (a character moves, and the camera moves with him).
Then there are the directors who like to plan big, technically difficult camera movements that require elaborate rehearsal on the part of the technicians and the actors. Vincente Minnelli is probably the king of this type of director -- the turning-out-the-lights scene in Meet Me in St. Louis, where the camera moves around an entire house in one uninterrupted take, required an unbelievable amount of rehearsal for cameramen, lighting people, and actors alike -- though F.W. Murnau is usually credited as being the father of the camera movement movement, and Max Ophuls was the director whose use of camera movement was most influential on directors like Minnelli and Jacques Demy.
And then there are some directors who almost seem to have a patent on certain kinds of camera moves. Steven Spielberg has done a certain type of move so often -- which can roughly be described as the camera moving in on a character while also moving down, so the character somehow seems increased in stature -- that it is indelibly associated with him; in the pilot of Buffy the Vampire Slayer when he needed exactly that kind of movement in a shot of Buffy, Joss Whedon asked the pilot's director to "give me the Spielberg," and the director knew exactly what he meant without another word being said.