How do you use Cepheid Variable stars to determine distance from apparent brightness?

It would seem implausible to be able to use the changes in apparent brightness of any star to figure out its distance, but in fact with a bit of added information, this is exactly what you can do!

In 1912, Henrietta Leavitt at the Harvard College Observatory was studying the light curves of stars which show how, over the course of hours and days, how the light from some kinds of stars change by measurable amounts in a periodic manner. In particular, she was studying variable stars in the Large Magellanic Cloud which means that all of the stars had the same distance. After studying several dozen of these stars, she plotted their periods against their apparent magnitudes and discovered that there was a very simple linear relationship between their period and their apparent brightness. Since they were all at the same distance, this apparent brightness was in fact a measure of their absolute brightness so that the stars with the longer periods were intrinsically the brighter stars compared to the shorter period stars. However, because she did not know what the distance to the Large Magellanic Cloud was, she could not calibrate the method this way.

The calibration of the Cepheid period-luminosity relation was accomplished over time by measuring the distances to nearby Cepheid variable stars from some independent distance method, then determining what the true absolute magnitude of the star was. This, then, calibrated the period-luminosity relation by specifying the luminosity of just one Cepheid star at a known distance, and with a known period. This was done by Harvard astronomer Harlow Shapley in 1917. The first calibration actually turned out to be in error because there are, in fact, two different kinds of Cepheid variables with slightly different period-luminosity relations. The Classical Cepheids are of the so-called 'extreme Population I' stars and are the more luminous. Population II Cepheids are found in globular clusters and other old stellar systems with lower heavy element contents compared to Classical Cepheids.

Physically, what is going on here was explained by Sir Arthur Eddington, and has to do with the characteristic vibration modes of stars with different masses. More massive stars are intrinsically more luminous that less massive stars, so the dynamical time scales immediately are related to their masses and therefore their luminosities. The reasoning goes like this:

Kinetic Energy = Gravitational Potential Energy

1/2 m v^2     = 1/2 G M(star) m / r

then 
         V = square root( GM/r)

also     V = 2 pi R / Period

so
        Period is proportional to    square root ( G x Mass/Radius^3 )