This has been pretty well answered above, but here’s my take on an explanation in layman’s terms:
Say you have a bunch of objects whose positions are described in a 3-dimensional (x,y,z) coordinate system. One way that you can change to a new coordinate system is by rotating your axes. One of the key properties of these rotations is that all the objects have the same lengths in the new coordinate system as they did in the old coordinate system. Note that this isn’t true of other coordinate transformations such as stretching one of the axes.
Now, in special relativity that there is another kind of coordinate transformation called a “boost”, which corresponds to going from one reference frame to another. Basically, you’re changing your definition of what it means to be “at rest.” (i.e., zero speed). Einstein argued that the laws of physics should still be valid regardless of how we define “at rest”, and showed that in order for this to be true our definitions of lengths and time intervals have to change whenever we redefine “at rest”.
So unlike rotations, boosts don’t keep length (in the usual 3-dimensional sense) unchanged. But a mathematician named Minkowski (who happened to be a former professor of Einstein’s) realized that boosts (as well as rotations) do preserve a sort of four dimensional length involving three dimensions of space and one dimension of time. So you can treat boosts and (spatial) rotations on equal footing, as rotations in a kind of four-dimensional spacetime. (Einstein drew heavily on this notion of four-dimensional spacetime when he extended special relativity to the general theory of relativity.)
However, the formula for these four dimensional lengths (called “spacetime intervals”) is a little different than the ordinary distance formula.
Instead of L[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup]
we now have L[sup]2[/sup] = x[sup]2[/sup] + y[sup]2[/sup] + z[sup]2[/sup] - t[sup]2[/sup]
Note that t[sup]2[/sup] has a different sign than all the other dimensions. That’s what people mean when talking about time being an imaginary number – it’s square is negative instead of positive. (There should also be a factor of c in there – i.e., the speed of light – which can be thought of as a conversion factor to put time in the same units as distance.)
Long story short: Time is the “fourth dimension” in the sense that it’s convenient to do physics working in a four-dimensional “spacetime” – but the metric (i.e., distance formula) of this spacetime isn’t just the usual Euclidean metric with four coordinates instead of three – time is singled out by having a different sign. Getting back to the original question, I suppose time can be thought of as “a dimension in the geometric sense”, only it’s not Euclidean geometry you’re doing.