What is a "Three-Dimensional Maneuver"? Many times in Attack on Titan, the various soldiers mention something called "three-dimensional maneuvering", which appears to be an effective strategy for the battles with the Titans. At the end of chapter 3, there is a "rough explanation" about it:



However, it doesn't really explain what three-dimensional maneuvering is, and the manga doesn't seem to show very well what the soldiers are doing when they perform these maneuvers. Logic would dictate that a three-dimensional maneuver can be any maneuver since movement in the real world is in three dimensions. However, this term seems to mean something specific.

What is a "three-dimensional maneuver", and how is it special or effective against the Titans?
According to what is shown in Episode 3, Three-Dimensional Maneuvering is indeed being able to move in all three axes. We, normally, are able to move two-dimensionally, meaning that we move along a horizontal plane. The gear introduces a vertical axis, making it possible for humans to move to a desired height as well.  



As to the advantages it might provide in fighting Titans, besides the ones Rinzwind says in his answer (avoid falling rubble and the like; and also, to make people move a lot faster), I guess the most evident one would be the fact that people do not need to fight at a height that only reaches a Titan's feet or legs and being able to rise enough to be able to attack them where it might do some actual damage. From the Attacks on Titan episodes I've seen so far (3 episodes), humans always go for the neck, meaning that this gear allows them to strike on more (presumably) vulnerable spots.