Actually I'm learning String Theory, and one of its proposals is that there are actually 25+1 dimensions of which only 3+1 are visible to us-- and the remaining are curled up. However, superstring theory says that there are 9+1 dimensions; and M theory says 10+1 dimensions.
I'm having trouble imagining "curled up dimensions"
Is it that we go out of the universe to see a 4th dimension? (This was suggested by a friend of mine)
Annav gave the correct answer, but here's some help on visualisation.
First thing: We cannot imagine more than 3 space dimensions. You can try, and get tantalizingly close, but it's extremely hard to wrap one's brain around it Due to this, I shall explain this in lower dimensions,and you can try to generalise it. Try.
Alright. Let's imagine a thin hose. You're standing nearby, and to you, the hose looks like a line. One-dimensional. On the other hand, an ant on the hose notices that it can move along two dimensions--along the hose and around the hose. In this way, the 2nd dimension is "curled up" and hidden except at small levels.
For a 2D example, consider an extremely thin sheet. This sheet is made of layers of mesh. For an external observer, the system looks two dimensional. But, an ant can move along the two dimensions of the sheet, and also perpendicular to it (up/down sheets). You can also look at this as a thin film of water and a tiny fish inside it.
We can't really extend this further than that. We can look at projections (like the Calabi-Yau manifold linked in Anna V's answer)--but these are like a cross-section. Not the whole thing..
We go out of the universe to see a 4th dimension--Well, this is sort of correct. Let's take a 2D ant living on a 2D scrap of paper. It perceives that there are two dimensions, and it is restricted to moving along these two only. If, by some unknown force(your hand), the ant is picked up, it will have moved through the third dimension. But, being a 2D ant, the ant will still "see" only two dimensions. The important part here is that something happened that could not have happened if the universe was restricted to 2D--a smart ant would reason thus "I can only see two dimensions, but something picked me up; thus there are more than two dimensions"
Similarly with humans. If we "exited" the 3D universe in that sideways manner, we would be moving through another dimension--but we would still see only three dimensions. But we can reason that more dimensions exist from this.
Here is a nice illustration of the Calabi Yau manifold. One can visualize at each point of our 3 dimensional space as a tiny manifold like that which encloses the extra dimensions.
Alternatively: if our third dimension were curled up we would be living in Flatland , without knowledge of the third dimension. One can rotate a two dimensional figure into the third dimension in our world, since the third dimension is not curled up. If it were as in Flatland, then the figure would not fit into the third, curled, dimension, because of dimensional incompatibility. Thus the flatlander cannot interact/see the third curled dimension.
