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I couldn't possibly explain it as well as this. This is based on String Theory, and explains all of the dimensions up to the tenth. Enjoy.

http://www.tenthdimension.com/medialinks.php

Edit:
This logic is very strange and confusing, and is not widely accepted fact among scientists. I wonder why...

Personally, I think it's just a loop of insanity. There are flaws in the overall logic of it, I admit, but there are also windows of odd logic that are opened when you talk of higher dimensions. Anything could make sense.

(And I don't agree with everything in that video. I don't think there should be just ten, and I don't think you can just *pop* places, and I don't think dimensions are all dumb lines. It's just food for thought... not propoganda.)

That video is a farce, a terrible, abhorrent mockery of mathematics and physics and even the basic tenets of rational thought. There is absolutely nothing of value in that video, and what it purports as facts and theories are nothing of the sort. It is lies and vile distortions, serving no purpose except to deceive, to mislead and misinform, to spread falsehood and unthought. I wish with full sincerity that that video and the creator of that video did not exist, and that it never existed, and that those who apparently tried and failed to educate that person also never existed, because there is no reasonable standard by which that video does anything but damage and corrosion.

That video is a sickness, a blight. Don't watch it, don't think about it, purge it from your thoughts and from your memories. It is worse than ignorance, worse than a lack of knowledge. It is anathema.

No, don't watch it. It's not even wrong in a "these are some of the usual mistakes to look out for" sort of way.
Creationist arguments, for example, would be the kind of thing at least worth skimming through, because there are people who will try to use these arguments in debate and it's always a good idea to know how the other side thinks.
The tenth dimension video is just garbage and fortunately doesn't have a lot of supporters, so you won't see anyone actually trying to quote it.

I'm critical of misrepresentations of the things I know about. That includes math, physics, artificial intelligence, and linguistics. It's one thing to argue philosophy or art, but these subjects have correct answers and pieces of drivel like that video are far from correct.

In physics, like in math, a dimension is a direction that can be used to describe a location in the universe without regard to what exists at that point. A dimension is just an orientation and a map to the number line, saying "In this direction, this point is a distance 1 from the center, this point is a distance 2 from the center, etc".
In other words, a set of dimensions allows you to describe a bunch of points in terms of numbers. Points on a two-dimensional plane are described by pairs of numbers, points in a three-dimensional volume by three numbers, etc.

On a map we have latitude and longitude; these are dimensions, although they don't quite match the Earth, but they still give us a way to describe a location on the Earth without having to know what exists at that location; we could describe the location of a city such that someone could find the city, without having to describe the city itself. And note that we have lines of longitude and lines of latitude; each line is a mapping from the globe to the number line.

In general, there are three physical dimensions that we can directly observe, i.e. the three spatial dimensions. Note that the three spatial dimensions act essentially the same way. If there's a big mass below you, you get pulled downward. If the big mass were to the left of you, you'd get pulled to the left. It's all the same.

There is a fourth dimension that we experience, time. This one acts differently from the spatial dimensions, for various reasons, but it is still a dimension, and with a certain viewpoint (relativity) it acts similar enough to the first three to be integrated with them, into "spacetime".

As far as we know, these dimensions are large enough for us to notice different points along them. We can look at two objects and say "this object is to the left of that object" or "this object is above that object".

String theory posits at least six more spatial dimensions. These six dimensions are so small that we can't observe them using the physics that we know of. Below a certain length, quantum mechanics states that it is impossible to observe things, and thus since these dimensions are each shorter than this length, we can't see them or different points along them. But they exist (according to string theory) and they can be moved along, just like the three spatial dimensions that we can see.
Even though we can't see them, string theory requires them. In string theory, particles are actually tiny vibrating strings, and these strings need a space to vibrate in; in order to be stable, i.e. not fall apart due to various mathematical considerations, this space needs to have nine spatial dimensions, and six of them need to be curled up into a what is called a Calabi-Yau manifold. Never mind what a Calabi-Yau manifold is; just note that it's complicated and the dimensions twist around each other and can't be separated like the dimensions of the plane can be separated into "up-down" and "left-right".
Keep in mind is that these six dimensions are identical, as far as we know, in that they all act exactly the same, they just point in different directions. None of this "this one is curling, this one is probability, this one is infinity" junk. A dimension is a dimension; the only difference is which way it points and how it interacts with the other dimensions.

This made me laugh inside.

In physics, a dimension is basically just like a dimension in geometry. A line is one dimensional in both math and physics, a plane is two dimensional in both math and physics, a volume is three dimensional in both math and physics, and so on.
Qualities of matter are usually considered not so much dimensions in physics but "degrees of freedom". "Dimension" in physics is reserved for space and time, i.e. location information independent of matter.

That's true about the phase space, although I've always thought of the phase space as more of a mathematical construct rather than an object of physics, just as the Hilbert space of quantum mechanics has an infinite number of dimensions, but those aren't generally considered to be physical dimensions.

And while mathematics in general has any number of definitions of dimension, geometry is rather picky about them; the only case where geometry has a multiplicity of non-equivalent definitions of dimension is with fractals, and since physics has not exhibited a fractal system yet, there is no real need to differentiate between the different notions.

Ohhh.... I always considered dimensions to be a quality of something. You know, like length, width, height, duration, and so on. sweatdrop I guess the definition is kind of flexible.