Curves are a mystery

It's easy enough to use a curve function (cubic, catmul, whatever) to draw a curve in 2D or 3D, but what baffles my mind is how to interpolate a curve over time.

Say you use a curve to interpolate between several values over time. These values could be keyframes, or gradient stops, or any value that takes place over time in a non regular interval.

I think the problem is sometimes phrased something like "how to interpolate y from x" where x is time and y is a set of data at non regular intervals