Imagine a closed universe.
For example you zoom off in one direction and end up where you started. You can do this in all 3 directions - a bit like the old asteroids game.
Now asteroids was played on a 2-d screen - how would the asteroids universe look?
Well if you think about it, it would be a torus, a bagel shape.
So for a closed 3-d universe you'd need a 3d hyper-torus.
What would that look like? well it's pretty hard to imagine but you can draw a cube on a piece of paper can't you? so in the same way we can construct a 3d projection of a 4d torus
There's some nice animations and descriptions of 4d opbjects here:
http://en.wikipedia.org/wiki/4-D_geometry
As for higher physical dimentions, as fo3nix says a number of string theories have been developed that require 10 physical dimentions - but there was no way to tell which if any were correct. Then a number od physicists including Ed Witten pointed out that if you assumed 11 dimentions all the seperate theories became aspects of a single theory which he named M theory.
The real challenge is to take this and force a prediction out of it that can actually be measured - until then it's just something to amuse theoretical physicists