Imagine a point - a singularity. That's the 0th dimension. Only one point exists. Everything within this universe exists at exactly the same place.

Now run a coordinate axis through the point. Suddenly you've got a line. Everything within this coordinate system has a single coordinate - we'll call it the x coordinate, and we'll call the axis the x axis. That's the 1st dimension. You probably know it as a number line from algebra.

Now run a new axis perpendicular to the x axis. We'll call it the y axis. Now everything addressable has two coordinates, an x and a y coordinate. This is 2-space, a plane, the 2nd dimension. We can put points anywhere on this infinite plane.

Now we're going to run an axis perendicular to the x and y axes. We'll call this new axis z. We now have 3-space, the 3rd dimension. We can address points in space - we can describe a cube or a sphere.

Now we will induce a mindfuck. Run a fourth axis perpendicular to your x, y, and z axes. We'll call this the w axis. Impossible, you say? Can't find a directon that will place this new line perpendicular to all the others? That's because you probably cannot conceive of this direction. You can't move in it, you can't see it, and you probably can't think of it in anything but an abstract way because you have no experience with it. Depending on which physicists you listen to, this dimension is either too damn small (see string theory), only gravity propagates in its direction (will try to find reference), or it doesn't physically exist. This is the fourth dimension. It may or may not have physical relevance, but mathematically it's completely valid, and even necessary for modelling certain things in 3-space such as temparature in a solid (anything that's a function of 3 variables can describe a four-space curve).

You can continue adding as many new dimensions as you please, without compromising validity. AFAIK the greatest number of physical dimensions that have been seriously speculated is seven (again, they are too damn small for you to interact with), but the magic of math lets you deal with n-space. Mathematically, adding more dimenstions is sufficiently trivial that textbooks may introduce extra dimensions without fanfare.

See also: Flatland, Flatland: A Romance of Many Dimensions, vector calculus