Do we really live in three dimensions? The recent hype of 3D movies in Hollywood seems to emphasize that fact, but Hollywood does not always get things right... There are two main reasons why you would want to consider this possibility. First, if you allow for more than three dimensions of space, interesting things happen in physics: a unification of certain physics laws that you may otherwise think are very different. The second reason comes from String Theory: this new framework of theoretical physics is the only consistent way we currently know for marrying quantum mechanics and gravity, and it predicts the existence of higher dimensions...
What does it mean to be living in higher dimensions; say four dimensions for fun. Imagine first a simpler case. An ant lives its miserable and monotonic life on a cylinder. It can move in merely two dimensions. But one of the directions it can move in is very different from the other: if it goes around the cylinder, it comes back to the same point; but if it walks along the cylinder, in principle it can walk forever (as always, cylinders in physics are infinite...). You, on the other hand, are a larger creature. You look at this picture from very far away. You just see a wire; you cannot resolve the thickness of the cylinder. To you, the ant lives in one dimension! In this manner, we could be living in four dimensions, three of which are very large or even infinite, but the fourth is curled up in a circle so tiny that we cannot resolve it. That is all nice and well, and would make for a great conversation topic in a bar, but does it buy you anything new in terms of understanding Nature?
The answer is a big yes. Imagine we are tracking an electron from our pitiful three dimensional perspective. From the real four dimensional perspective, the electron may move around in the fourth dimension that we cannot resolve - just like the ant circling the cylinder. An electron has an attribute called electric charge. It is the reason it feels an electric force. But this quantity called charge is rather mysterious: what is it? where does it come from? From the viewpoint of the four dimensional world, electric charge is simply a measure of the speed of the electron in the fourth dimension! And the electric force is a part of the four dimensional gravitation force! When you cannot resolve the tiny size of the fourth dimension, you instead see two forces: gravity and electric force... but they may be the same thing in reality. So, despite not being able to see the fourth dimension, we can still see its effects dramatically from our three dimensional perspective. Hence, knowing about a fourth dimension can give us deep insight about how Nature ticks.
In String Theory, higher dimensions are a must. In achieving the long anticipated marriage of quantum mechanics and gravitation, String Theory needs a series of logical consistency checks. One of these requires that the world is higher dimensional, 10 space dimensions… That means 7 of the additional dimensions must be curled up into small sizes and we are thus unable to resolve them. But their existence provides a myriad of indirect effects on our world, effects that unify the natural laws and the spectrum of subatomic particles in elegant ways. There are currently several experiments that are trying to "see" or "resolve" extra dimensions. Depending on how small these extra dimensions are, and how intoxicated the experimental physicists conducting the experiments are, we may sooner or later find out we need a new pair of dorky eyeglasses to go to the next blockbuster flick.