What the heck is "time dilation?"

This explanation and the YouTube illustration should help -

Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock. This is often interpreted as time "slowing down" for the other clock, but that is only true in the context of the observer's frame of reference. Locally (i.e., from the perspective of any observer within the same frame of reference, without reference to another frame of reference), time always passes at the same rate. The time dilation phenomenon applies to any process that manifests change over time.

In Albert Einstein's theories of relativity, time dilation is manifested in two circumstances:

In special relativity, clocks that are moving with respect to an inertial system of observation are measured to be running slower. This effect is described precisely by the Lorentz transformation.

In general relativity, clocks at lower potentials in a gravitational field — such as in proximity to a planet — are found to be running slower. This gravitational time dilation is only briefly mentioned in this article; see that article (and also gravitational red shift) for a more detailed discussion.

In special relativity, the time dilation effect is reciprocal: as observed from the point of view of any two clocks which are in motion with respect to each other, it will be the other party's clock that is time dilated. (This presumes that the relative motion of both parties is uniform; that is, they do not accelerate with respect to one another during the course of the observations.)

In contrast, gravitational time dilation (as treated in general relativity) is not reciprocal: an observer at the top of a tower will observe that clocks at ground level tick slower, and observers on the ground will agree. Thus gravitational time dilation is agreed upon by all observers, independent of their altitude.