Finite means that it can be measured using a real number.
Unbounded means that it has no "end".
Normally these two words are contradictory.
However, in geometry, they can be used separately.
Take Earth's surface: it can be measured with a real number.
Knowing Earth's radius (a finite number), we can measure Earth's surface area. It is finite.
However, if you start off in a given direction, you will never reach the end. In the 2 dimensions of the surface, you can travel in one direction and never reach the end. It is unbounded. There is no real number that can represent the maximum number of kilometres (or miles) that you can travel in one direction.
Mind you, you will pass over your starting point over and over again, but that is because the surface is finite.
Some people think that space is a similar arrangement, except that it is a 3-dimension volume, "wrapped" around a finite radius in a 4th dimension. You start off in any direction, you will never find an end. However, if you could travel long enough in that direction (without turning around) and if it were not for expansion, you would eventually pass your starting point.
Others believe that the universe is really infinite in spatial extent (it extents forever in any given direction. Of course, if it is truly infinite, then it is also unbounded (there is no end).
The word "unbounded" also has other applications which are similar -- but not quite the same.
Two objects are "bound" if they are held together by gravity. Earth is in a "bound" orbit around the Sun. The Voyager probes are on "unbounded" orbits and will (eventually) leave the solar system, never to come back.
The term can also be used to describe the density of the universe: is there enough matter in the universe, so that gravity will eventually win and cause expansion to stop? If yes, then the universe would be "bounded" by gravity. However, for now, it appears "unbounded": expansion would continue forever.