Select the Move Tool, just to the right of it (on the main toolbar) is a drop-down list which probably says "View". Expand that list and select Local.
Note. By default the 3 Transform tools (Move/Rotate/Scale) each have their own Coordinate Systems - switching between the tools also switches the Coordinate System. This makes using the tools much more flexible, but can also be confusing when you're not expecting it. This is also true at sub-object levels, but to make things even more confusing they are separate from those used at the Object level.
I would play with the possibilites in a test scene to become familiar with them, and remember that the Coordinates Center button (next to the drop-down list) also affects the way these tools work (especially Rotation).
Max 2016 (SP1/EXT1)
Win7Pro x64 (SP1). i5-3570K @ 4.4GHz, 8Gb Ram, DX11.
nVidia GTX760 (2GB) (Driver 430.86).