Imagine the inhabitants of a two-dimensional world, something like Edwin A. Abbott's Flatland. In this land, there is no such thing as height; all of the universe is contained in the plane, and there are only two directions of motion: forward or backward, left or right. Suppose the inhabitants of this world discover that there are regions in their space where it is more difficult to move than others. Furthermore, they discover that each object creates such a region, with a size proportional to its mass. They find, therefore, that when two objects are close together, it is harder for them to move apart then if they are separated by a great distance, as if they are attracted to one another. How could the residents of this land explain such a thing? They might say that there is a mysterious attractive "force" which acts upon all things in their universe. They would probably make up long equations and laws to describe and predict the consequences of this "force." They might even give it a name, say... "gravity." But have they really explained anything?
Now imagine that, one day as you descend into your basement, you come across this flat world. You see thousands of little creatures moving about, bumping into one another, and otherwise carrying on. Curious, you poke your finger down into this world, creating a deep depression. From all directions, the little people fall helplessly into the well you have created, screaming that a giant planet (your finger, which to them appears as a circle) has appeared and is sucking them all in with its immense gravitational force. From your perspective in the third dimension, however, there is no mysterious force. The little people are merely experiencing the effects of curved space. It is more difficult to move in a curved region in space, as anyone who has climbed College Hill knows.
Now, stretch your imagination to suppose that our own four-dimensional universe is likewise curved. Gravity would, as for the flat-world residents, be merely the effects of curved space. Albert Einstein imagined a world precisely this way, and in 1916 published his theory of General Relativity, which proposes that space-time is warped and curved by matter, which appears to us as gravity.
Building upon Einstein's work, physicists began to question whether we might be just like the Flatlanders, living in a universe of many dimensions, but aware of only a few. They began to discover that, by allowing their theories to include more and more dimensions, they could account for all observable forces. The most popular of these theories, called Hyperspace theory, requires a ten-dimensional universe.
Of course, there is no definite way to know exactly how many dimensions our universe has. Like the Flatlanders, we can only find hints of what lies beyond our four-dimensional perception. Our universe may be contained within a reality of thousands of dimensions, or of infinite dimensions. Unless those dimensions interact with our four (or ten) in specific ways, there is no way that we can detect them. Not until their inhabits come to visit, anyway!