Here is a Flash animation that explains the ten dimensions of string theory. (Thanks to Alex Tabarrok of Marginal Revolution for the pointer.) Don't go away, it's really worth a few minutes of your time, as a mental exercise, even if you have no interest in physics. To help you through the rough spots, I've concocted this summary:
0. The zeroeth dimension is a point, an abstraction of the postion of an object in a system.
1. A line joining two points forms the first dimension. A line has length, but no width or depth.
2. A second line branching off the first adds a second dimension. We now have length and width, but not depth.
3. A third dimension results when the second line is "folded" back on first, enabling movement between the two branches (length and width). Thus we now have length, width, and depth. (Think of a flat piece of paper that is rolled into a tube, so that a point on one edge becomes adjacent to a point that had been on the opposite edge.)
5. The multitude of paths that objects could follow through time (according to quantum mechanics) are branches from the time line. These possible paths constitute the fifth dimension.
6. A sixth dimension results from the folding of the multitude of paths, so that an object can jump from one possible future state (path) to another. The collection of all such possible moves is a point.
7. Each point in the sixth dimension represents all possible outcomes, through all of time, for a given set of initial conditions and physical laws (e.g., the speed of light). The seventh dimension is a line that joins all such possible points, representing all possible initial conditions and physical laws.
8. The eighth dimension is represented by branches from the the seventh-dimensional line. The eighth dimension is analagous to fifth dimension. That is, it represents all the possible "universes" that might result from each of the possible starting points as they move through time.
9. The ninth dimension is analagous to the sixth. That is, it represents the movement from one of the possible "worlds" to another because of the folding of the eighth dimension.
10. The tenth dimension is a single point that encompasses all the possibilities inherent in the ninth dimension. It is the ultimate dimension because, by definition, it encompasses all possible worlds and all of time.
Only the first three dimensions seem "real" to the typical person, who observes the world unaided by scientific instrumentation and theories. Einstein's special and general theories of relativity account for the interactions of space and time (the first four dimensions). (For an accessible explanation of the special theory, read Lewis Carroll Epstein's Relativity Visualized.)
Everything from the fifth dimension onward seems to hinge on the controversial "many worlds intepretation" of quantum mechanics. Quantum mechanics lays the foundation for a fifth dimension, but in a tacked-on way. Some 80 years have passed since quantum mechanics became the accepted view of physical behavior at the sub-atomic level, but there still is no generally accepted unifying theory for quantum mechanics and general relativity (see quantum gravity), let alone a "theory of everything," of which string theory is one example.
In sum, everything from the fifth dimension onward falls in the realm of scientific speculation. Science proceeds from speculation based on observation, but speculation should not be mistaken for scientific knowledge.
In any event, enjoy the animation.