What exactly is a dimension? Nobody really knows for sure, the people with the best theories though are theoretical pysisists, so I asked them the question. Here are a selection of their replies on the question - What is a dimension.

There is a theory of strings
supposed to unite the 4 forces of nature because physicists have found
stunning implications of the theory such as that the theory includes the
graviton; the quantum of gravitation. The theory only works with
more dimensions then we presently see. Now the mathematics is very
hard and I won't go into too much detail but scientists have used the Kaluza-Klein
geometry and stated that the reason we cannot see these other 5+ dimensions
is because they are rolled up into extremely tiny spaces; Planck scales.
It is also very hard to talk about extra dimensions because we cannot describe
them in any way because we in everyday life deal with only 3 spatial dimensions
and 1 temporal.

Jason, The classical dimension has the property of independence. Given a particular variable, any value such variable acquires is not visible to (does not carry over, does not influence) any other variable. In a classical example, objects on the floor (in a plane) will not come into contact if these move anywhere in the ‘z’ (up/down) dimension. You can see that time has nothing to do with it since if x, y, and z are the same, a collision will occur regardless of time (regardless of when) and independence no longer exists.

In a computer it is trivial to have many degrees on independence (many dimensions). This is because data can be deposited in memory and such data does not need to relate to anything else and can change without any other consideration.

There was a very good post earlier (which after a short search I could not find) about using a dimension as another determinant of normalcy. (x, y, and z of spatial distance are independent if they are normal to each other.) This is an excellent way of looking at dimensionality. The example author gave was the dimension that separates the kinetic and the potential energy of a pendulum. Kinetic and potential energy are not independent of each other. In fact, they depend on each other. Yet, the normalcy dimension shows the balance (or “path of neutrality”) between the two energies as these transform into each other in the pendulum example. In other words, a dimension is not necessarily restricted to spatial distance and this is also the concept behind (my own) stuff on HyperFlight.

Sure you want to know more?
There is no object in the universe that is truly independent. There
is no property of an object that is truly independent. Spatial distance
is just one such property that can be (but is not always) independent.
Esoteric? Not if it leads to practical and observable results.
As for literature, Pythagoras is a good start (it’s been only 2,600 years
ago.) You may want to throw in computability for good measure if,
for example, you want to understand why it is “Four horsemen of the apocalypse,”
rather than three.

I think that this extra dimension
has not any simple signification that the usual. For example we can
easily explain to anyone why live in a universe of 4 D, for example if
you want to met with somebody you have to choose a place in space (3 D)
and a time(1D). The reason of why we use universes of 10 dimensions
is more mathematical than the others. In fact is a strange thing
and string theories already have to explain why we can't see this extra
dimensions.

Now I'll try to explain
what are this mathematical reasons.

We have GR, this is a theory very powerful that explain the gravitatory forces. But this theory have a very important property, in this theory 'gravitatory forces' really don’t exist. How is it done?.

Ok Imagine that you are a 2D animal, and you walk in the surface of an sphere. Now imagine that I see you from the outside of the sphere, and that this one is invisible to me. Then I'll see you travelling around, and that you don’t go straight. In order to explain that I imagine that some 'force' is making you turn, but in spite of this force we can see that is the space that is curved. That is what GR makes, a mass bend the space, and then the particles that travel in it don’t goes in straight lines, there don’t exist gravitational forces.

Now imagine the case of an electric field, ok one may think that it is easy to make the same as in the gravitational field, the electric field makes the space bended. But we have a problem now, that is that not all the particles can feel the electric field, only those who are charged. This make the concept of curved space inappropriate for the electric field, just because the curvature depends of the electric charge, and this is very difficult to imagine, the space is one, not one for each charge.

To solve this and unificate all the four forces in one scheme physics had tried a lot of solutions, we can for example make the space not only to curve, but also to 'turn', and a lot of different things, but none of this gives a satisfactory solution.

One a physic try to make
the next, in spite of trying to increase the degrees of freedom of space
making it 'turn' and all this, he try to add an extra dimension.
When this is done, we can add the electromagnetic interactions (of course
only classical) to the scheme of a curved space, but now in 5 dimension.
All the problems are solved, with this, we can explain how charged particles
travel in our universe. This is called the Kaluza Klein theory.
This was done a lot of time ago, now we know that classical electromagnetic
interactions are only an approximation of Quantum electrodinamics.

But the trick of add more
dimensions to our space time has seemed to be very powerful, and now we
have string theories that try to unify the four fundamental forces.
But, where are this extra dimensions?, What really means?.
No one knows this answers, we only know that in 10D physics seems more
easy. To explain why we don’t see this extra dimensions in our world,
Physics make this extra dimensions to roll up to very, very short distances
(of the order of the plank scale).

But all this is only theory, there are a lot of things that are not well understood, for example why 10 dimensions and not 234?, Why you roll up just 6 dimensions of the 10 and not 7, or 3?, What does this extra dimensions means?, Maybe the extra dimensions represents the 'state' of the particle.

Ok I have tried to show you why 10 dimensions are used in physics, but really there are only mathematical arguments. In my opinion the physics of string theory are far of us. When Einstein imagine our universe as a 4D one he has very good physical reasons, but with the others we cant say the same...

As you have seen there is
a lot of my personal point of view in this, but More or less the things
are this. I’m not trying to say that string theory in false or things
like that, the fact that we can unify gravitation with this theories, and
no infinities arise means that there are in the right way, but I think
that we don’t understand his physical significance (i.e. We have to discover
the equivalent to the principle of equivalence to the GR).

As others have talked about, 10-D theories arise in some string models. I will talk instead about the idea of a dimension generally and at least one way to translate that idea to physics intuition.

Our initial idea of a dimension certainly comes from the space we ordinarily experience, with its three directions. But why do we say there are three such directions? Well, because they are independent and "orthogonal" ways that something (a position) can vary. What do I mean by "orthogonal"? Naively that means "at right angles" and just refers back to the same space-image. But a more generalized sense of it is defined mathematically, and is familiar enough in such fields as statistics and/or linear algebra.

This more generalized sense of orthogonal might be described as follows. If we consider any two things we can measure and a bunch of data, we may see the data as correlated. Whenever we do, we can construct an idea of "one direction" in which the correlated things "are similar" and "another direction" in which they "are different". Say we are just looking at the heights and weights of a bunch of people. The results will be correlated. We can reduce that part of the data to people's position along one height-n-weight scale, as a line through the data (call it, for no good reason, simply "size"). But another part of the data will not be correlated, and would show up in a plot as items distant from the line we just drew. We could plot that too for each (call it "obesity", perhaps).

What have we done in the above case? By looking at the data a bit differently, we have taken out the correlation between measured things and replaced it with numbers put on things we didn't directly measure, which are now *not* correlated, or are independent. We say we have reduced the data to a orthogonal "basis", in the sense of a set of "axes" in a way intrinsic to the data themselves (rather than merely to our initial and perhaps silly ideas of what to measure). In that example, we started with two measured things and ended with two measuring things - the number of "dimensions" in the data stayed the same. But in more complicated cases, we might be able to reduce some picture's complexity quite a bit in that fashion.

So, that is an idea of "orthogonal" that is independent of our mental images, that we use in statistics commonly enough. Generalizing that procedure, people often analyse some practical, complicated engineering or prediction problem (in economics for example) to a "linear algebra" model. Meaning, they look at the system they are studying as being n equations in m unknowns; especially nice are cases when n and m are the same, because then we know those can be solved by substituting the results of one into another.

But whether solveable that way or not, the same idea of taking the data and "lining up" correlated parts, and labeling that the first direction, then taking the remaining variation and finding what is correlated in that, etc, can still be used. The result is a "basis" for the data, a set of "directions" that needn't really be "directions" at all but are more like "independent aspects" of something being looked at.

So that is the mathematical idea behind using many dimensions. But what is the physical intuition we can use to connect that generalized idea with uses of many dimensions in physics? To me the most useful idea here, which I personally find quite helpful, is to apply this sort of thinking to what is called the "phase space" of some physical system. You may already be familiar with that from classical physics, but in any event I will briefly explain the idea.

A phase space is a different way of looking at some physical system, in which each point in the space represents a possible entire state of the whole system. Paths in phase space then represent transitions from one state to another, or possible "evolutions" of the system. This is very clear in a simple, classical physics example with only a limited number of things going on. You can often specify such a system with a plot of just the potential energy on the one hand, and the kinetic energy on the other. Everything else about the state follows from what the system is or how it is set up, say (e.g. a simple pendulum).

Then you look at what goes on as moving from one "state" or "phase" to another, as potential energy becomes kinetic ("go here" in phase space), then potential again ("go back there" in phase space). Nice, clear idea to look at classical systems with.

And the neat thing is that physical principles like "least action" can be applied directly in the phase space, without needing to track back to ordinary dimensions and translate. So the "calculus of variations" and such can be used in the nice, simple phase space (with only its kinetic and potential "basis" "directions"), and will still specify the path from one system state to another. If you want ordinary coordinate results, then after you have solved the whole thing you can translate back from phase space to normal space at the last step.

So, the mathematical idea of a dimension is more like "uncorrelated way something can vary". A physical. intuitive application of that more generalized idea of dimension in even classical physics is the notion of phase space for some system or other.

When string theorists put theories together in 10 dimensions, you might think of them as working in a phase space that specifies all the uncorrelated, independent ways something can physically change or vary (including charge, "QCD" color, etc.). Now, as others have pointed out, what sense to make of those theories and the varying things they posit in this or that model, is a different question, as is the question whether those theories are any good or not.

But perhaps the above may
help think about what they mean by additional dimensions.