Actually the term two dimensional doesn't tell the whole story here. The surface itself is actually two dimensional -- as you walk along the surface, your choices can be described with two directions -- east-west and north-south. But the surface is living in a three dimensional world. As you walk along the surface you will go in the up-down direction as well. We say this surface itself is intrinsically two dimensional but extrinsically it is living in a three dimensional world.