Most of our understanding of the Universe is via Newtonian mechanics and involves the three spatial dimensions up/ down; left/ right; forward/backwards.
And Time.
All of Newtonian mechanics and the large scale physical universe only require these four dimensions.
But what about the small scale? The Quantum universe of electrons and photons and leptons and so-ons.
Sorry :)
Here's a fun thing: http://www.youtube.com/watch?v=ViQoUXu5uK0

Taking on board @Meitnerium278's cricket ball analogy: where the surface of the cricket ball is now the surface of a rubber balloon that is in the process of being blown up. An ant on the surface of the inner surface of the balloon would see all points on (in) that surface receding away from her: Equally. Now imagine that instead of an ant and a surface that the the rubber fabric of the balloon is our three dimensional universe (plus time!) and the ant: us. The points are the receding galaxies at the limits of our telescopes.

But what if the rubber fabric of space-time were folded.
http://www.moillusions.com/2006/10/eschers-moebius-ring-with-ants.html
Thus our two dimensional ant living on its two dimensional treadmill, actually moves through three dimensions (plus time) without even realising it!

Multiple dimensions. The latest in handwaving (Theoretical Physics) seems to involve eleven physical dimensions but these are teeny tiny little dimensions that are impossible to see or measure but are needed for some very speculative thinking and the mathematics to back it up.

Personally I am of the opinion that just as we have three spatial dimensions we also have three temporal dimensions. Somewhat further along these lines.
http://www.physorg.com/news98468776.html
Only more so.

Then of course there is the purely mathematical construct of Hilbert space that has some eminently practical applications and some fun paradoxes

http://en.wikipedia.org/wiki/Hilbert_space and his Infinite Hotel

http://www.bbc.co.uk/dna/h2g2/A4080467

There was a somewhat excellent TV Movie http://www.imdb.com/title/tt0443537/
*And Oh joy it's on the InterTubes*
http://vimeo.com/9016217
I last saw this an infinitely long time ago...
Thanks for making me dig.

"The important thing is not to stop questioning. Curiosity has its own reason for existing. One cannot help but be in awe when he contemplates the mysteries of eternity, of life, of the marvelous structure of reality. It is enough if one tries merely to comprehend a little of this mystery every day. Never lose a holy curiosity."
Albert Einstein