Which brings up an interesting question: what is a dimension. We normally describe the three dimensions as length, width and height. This is utterly wrong.
Let's veer away from the fancy math and into Computer Science, specifically data warehouses. A data warehouse is a buzzword for a big ol' pile of data that you want to analyze. When we have a big pile of information we want to go through, we need a good way to organize it. The common approach for a data warehouse is the dimensional approach. Let's say you had a big pile of sales data you wanted to crank through. We'd call each sale a "fact". And then, each bit of information we want to know about a sale, like what item(s) it was for, where it was shipped to, what credit card was used to buy it- these are our "dimensions".
That choice of terminology matters. We have facts (our things) and dimensions (properties of those things). Now, a customer calls up and wants information about a sale. They don't remember their order number (which would be your unique identifier and make this too easy).
So now we have a real question: how many properties of an order do you need to know in order to find it?* How many dimension do we need to use to find the sale the customer is talking about?
Now, let's go back to points in space. You all recall your grade-school graph paper with its Cartesian coordinate system. Your (x, y) coordinates. You had an arbitrary origin, and you would specify x, how many tick marks across and y, how many tick marks up or down a point was. (0,0) would be a point at the origin.
That's what we think of when we think of a two dimensional coordinate plane. Length and width. Convenient, and certainly, the way we're used to measuring things, but two dimensions is not just length and width.
The number of dimensions in a coordinate system is the number of properties required to uniquely identify a point in that coordinate system. A two-dimensional space is two-dimensional not because it measures length and width, but because it only requires two pieces of information to uniquely identify a single point in 2D space. Those two pieces of information don't have to be (x,y) coordinates, it could just as easily be (θ, d)- that's an angle and a distance. Think about bad war movies, where you hear something along the lines of "Hostiles at 3 o'clock, about 100 yards." Angle and distance. Two pieces of information.
So let's scale this up. 3D space. That means three pieces of information. We usually think of length, width, and height, but once again- it doesn't need to be. (θ, d, h) would work- angle, distance, and height. Or (θy, θe, d)- yaw angle, elevation angle, and distance.
We live in 4D space-time, so to absolutely position a point in our universe, we need at least four pieces of information- that fourth one is the dimension of time. When something happens.
I find the information theoretic approach to understanding dimensions powerfully enlightening. All of the sudden, Superstring theory makes more sense. For the unfamiliar, Superstring theory supposes that all matter is made of vibrating strings that vibrate in 10 dimensions. There's not a lot of evidence for this, but it's a pretty workable model, but it raises an interesting question- why don't we see 10 dimensions. Everything in our experience is organized by four pieces of information- we've never seen a point in space that needed ten pieces of information to describe it.
The "dumbed down" approach for laymen is to claim that the remaining six dimensions are curled up very small- so small that we could only find them by bashing particles together in a particle accelerator. When you take the information theoretic approach though, it does make sense. Those extra six pieces of information? They don't matter much at the scale we live in. Point (0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is, to our ability to detect, nearly identical to point (0, 0, 0, 0, 0, 0, 0, 0, 0, 1). Doesn't matter what units we're using, it's the relative sizes that matter. Differences in those extra six dimensions don't amount to much.
It's like the difference between painting a wall cream and painting a wall eggshell- yeah, there's a difference, and it matters to someone, but you're never going to be able to tell†.
Note, this doesn't explain or justify Superstring theory- it's still very obtuse and not well supported by evidence. The fact that its math requires extra dimensions that we don't normally interact with is also suspicious. But the whole thing is fascinating, and it's another way to get a handle on some of this higher dimension stuff.
*This generally isn't what data warehouses are used for, but it's helping me make my point, so bear with me.
†Yes, you can tell the difference between cream and eggshell. I know. You're very special, and always have the same lighting on hand.