A dimension is a measurement of something that changes. The something can be an object such as a rubber band, which expands and contracts through the dimension of length. It can be an extension ladder that grows or shrinks through the dimension of height. It can be an ice cream sandwich that loses the dimension of width as it is eaten.
How do we describe a multidimensional object? With coordinates.
For example: When a person walks down the sidewalk, he moves through three dimensions of space in a period of time (the fourth dimension). A mathematician would say that at any given moment the person has three coordinates of space: numbers corresponding to distance in terms of length, height, and width. And, of course, a time dimension coordinate is described as -- what time it is!
Nor are dimensions limited to space and time. Heating an object pushes it through a dimension of temperature. Weight is a dimension of mass.
When a mathematician wishes to move an object through more than the ordinary four dimensions, she simply adds coordinates for the extra dimensions. The four-dimension, space-time coordinates for the person on the sidewalk might be: a length of 20 feet (x = 20) from the beginning of the sidewalk; a width of 2 feet (y = 2) from the edge of the sidewalk; 6 feet (z = 6) above the walk's surface; at precisely 10:30 in the morning (t = 10:30). Add a coordinate for body temperature -- 98.6 degrees Fahrenheit, if the person's healthy (f = 98.6) -- and, voila, that person is described in five dimensions with coordinates x, y, z, t, and f. You can add on as many variable coordinates as you can dream up.
What about extra dimensions of space?
According to currently accepted theory, when the universe was born in a massive explosion -- the Big Bang -- all of space and time were created. Three dimensions of space and the dimension of time were very large -- that is, as large as the entire universe. But the seven extra dimensions described by M-theory did not explosively unfold. According to the theory, they remained tiny and curled up. And of course, only extremely tiny objects can be measured by the tiny dimensions. So a little object -- such as a string -- has three coordinates of large space; one coordinate of large time; and seven coordinates of tiny space. Each string can be described with a total of 11 coordinates.
Think of the seven infinitesimal spatial dimensions not so much as places as measurements useful for determining where a small object exists.
Do not try to visualize the space occupied by an 11-dimensional object in your brain, because if you try, you may well go crazy; except through higher mathematics, the human brain cannot generally conceptualize more than the four familiar dimensions our senses convey to us.
You insist on trying anyway? OK: Imagine that you are a Line-person from the land of Line. Naturally, you live on a line -- you are a one-dimensional person living comfortably in a one-dimensional space. What happens if a two-dimensional circle comes to visit? How do you see it? You see it from the side, as a line, of course. Being a one-dimensional person, you cannot visualize something in two dimensions -- such as a circle. If you want to "see" the circle, you have to evolve into a two-dimensional person.
Say you are happily living in two dimensions and a sphere comes to visit. What do you see? A circle, of course, because you cannot see depth. Since a sphere is three-dimensional, you cannot perceive all of it in two dimensions. You have to move up to the third dimension in order to see all of the sphere.
Physicists have discovered that by stepping into higher dimensions, they increase their perspective in relation to objects. A one-dimensional person stepping into two dimensions sees the whole circle, which, in three dimensions may turn out to be a sphere.
The same type of thinking has been followed to yield the 11-dimensional perspectives of string theory.
When physicists view the tiny loops of string in only four dimensions, they see only certain modes of the vibrating string, reflected as electrons, protons, photons, and other small, but measurable, entities of our physical world. When described mathematically in 11 dimensions, more of the string's existence, and more of the richness of the universe, is revealed.