Mark Turner:

A better explanation for time dilation doesn't involve this "one second per second" business.

It is a general observation that "moving clocks run slow". What that means is that if you compare the time measured by two identical clocks, one stationary relative to you and one that is moving relative to you, then the moving one will appear to be running slower than the stationary one.

The thing that strikes people as particularly puzzling about this, once they get over the notion that time doesn't run at the same rate for everybody, is that the same thing works in reverse. That is, if A and B both have identical clocks and are moving at a constant velocity relative to one another, then according to A, B's clock will be running slow, and according to B, A's clock will be running slow. Of course, if they ever get back together to compare the elapsed times on their respective clocks, they must both agree on the elapsed times on both clocks (which need not be the same). But the "getting back together" part will usually involve one or both of them accelerating, and accelerating can mean that the symmetry between the two observers is lost.