The accepted theory regards the space where we move in as a 3 dimensional surface in a quadri-dimensional universe. So, if you move towards one specific direction you will end up in the same place. If this sounds tricky, imagine that we are 2 dimensional creatures. The space where we move is then the surface of a 3D sphere. It's more less what happens in the surface of the earth.

When people say that the universe is expanding, what that means is that the radius of that 4 dim. sphere is increasing, thus it's surface increases too. So, imagine that the earth's radius is increasing. If you are in the equator and move to the east, for example, you will still arrive to the spot where you began, but the distance you have to cover is constantly increasing.

Offcourse, it's not that simple, because there is a lot of matter in the universe (some of which we can't detect), and according to Einstein's theory of general relativity, matter deforms the curvature of space. In the limit, you have a black hole, that leads you to God knows where.

I can't go much deeper, because I never had a course on Cosmology, but I hope this helps.