We only have proof of the 3+time that we all experience every day. However there are a number of theories ( String theory and M-theory amoung them ) which would neatly explain a number of things if you had many other dimentions.
The number of dimensions required varies between theories and even between variations going as high as 26 but 10 or 11 are commonly cited.
If we do have other dimentions the question arises of why we don't see them. One possibility is that they are very small.
Imagine if one of our 3 dimensions were only 3 meters high - throw up a ball and it would hit you in the bum! Now if these extra dimensions are like this but many times smaller than an atom we'd never know they were there.
So the real answer is "we don't know" and until someone comes up with a good experiment that we could try to find out all these extra dimension are really just something that theoretical physicists talk about in the pub when they think the experimentalists aren't listening