with regard to space, what does finite but unbounded mean?

Answer 1

It means that we can measure the size of the universe, but if you tried to approach an "edge" of the universe, all you would do would be to extend it. Wherever you go, the universe goes with you, so it is unbounded. You cannot go "outside" the universe.

Answer 2

Think of the surface of a sphere. The surface itself has no boundary and it is finite. In general, a finite unbounded surface usually requires some type of curvature (although there are tori that have zero curvature).

For three dimensions, there are similar constructs. Imagine that no matter which direction you go, you will eventually come back to your starting point. This is true for the surface of a sphere, but there is also a three dimensional version of the same thing (the sphere's surface is two dimensional).

The subject of three dimensional geometry and the nature of finite unbounded three dimensional figures is very complicated. The poincare conjecture was one problem that people tried to solve for about 100 years. It was just solved a couple years ago.

Be aware that there are also four dimensional versions that can be used for models of spacetime.

Answer 3

It means this: You've travelled a finite distance through space, but you keep going without end.

Example: I've travelled 100 miles, but I keep going forever. The distance I've travelled is finite, but the distance I can potentially travel is infinite.

Answer 4

means you've gone this far, but there's still more out there.