Current cutting edge physics talks about the universe being made of eleven or so dimensions, not the three physical and one temperal (x,y,z,t) of Einstein. This concept is very hard to visualize, of course.
One of the best books I have read that helped me get an intuitive grasp of the many dimension idea is Galileo's Finger, by Peter Atkins. I highly recommend the book for thinkers, for people who want to be taken past what they already know.
In my high school classroom, I marry an idea from his book with my own examples to show why extra dimensions arise in physics. In a nutshell:
1. In senior physics, students learn how to solve 2-D collisions. Ball A of a certain mass and velocity hits ball B, with its mass, and velocity, at some angle. The laws of conservation and momentum allow the two final velocities (including direction) to be calculated. Basically, two equations are used to calculate the two unknowns (the two final velocities) from the initial data.
2. What if there were three bodies hitting at EXACTLY the same time? Now there are three unknowns, but still only two equations. So, exact solution can't be calculated. This is the 3-body problem, still unsolved. (We can get approximate solutions by numerical methods, but the exact solution is not known.)
3. How about if one object comes in from the west and hits two identical objects balls lined up in contact north-south? (i.e. ball at 12 o'clock, ball at 6 o'clock touching in the centre of the clock. Other ball comes in from 9 o'clock) This 3-body problem is solvable because of the symmetry. The two stopped balls will pick up the same velociy, different directions but equal angles from 3 o'clock direction.
4. Thus if there is symmetry, we can calculate answers to problems which would otherwise be unsolveable.
5. Atkins points out that a line segment horizontal line segment is equivalent to a vertical line segment. We just rotate it (or turn our head!) Similarly a square and a diamond are equivalent, because one is a rotation of the other. (Turn the square 45 degrees.) Are a square and a hexagon equivalent? No. You can't rotate a square and make it a hexagon. Except that...
6. Draw a square and a cube. The cube is the 3-D object whose shadow is a square. But if you rotate a cube, its shadow can be a hexagon. (To see this, draw a box whose corner is rotated toward you, viewed from slightly above.) Therefore a hexagon and a square can be considered equivalent, or symmetrical, if you think in 3-D instead of 2-D.
7. We have already seen that if you find symmetries you can solve previously unsolveable problems. And we just saw that by viewing a concept from a higher dimension, you can find symmetries that were not apparent (or not there!) in lower dimensions.
8. Physicists are attempting to find formulas to describe the fundamental subatomic particles, and the behaviour of radiation and matter at the very small and large, and quantum theory, and relativity, and gravity (as discussed in general relativity). They haven't succeeded in finding exact solutions to the various equations that combine to describe things. BUT...
9. By working in more directions, symmetries were found that enable solutions.
10. Apparently, solutions exist if you use eleven dimensions.
So, there you are. Like all concepts in science, we find explanations that satisfy us and are useful. Eleven dimensions proves to be useful. Hence, proposing eleven dimensions becomes useful.