Cepheid Variable Stars?

The period of variability, which we can measure, is directly related to the absolute luminosity of the star. We can also measure the apparent luminosity of the star. The apparent luminosity is related to the absolute luminosity of the star by it's distance (raised to the 4th power, as I recall). Therefore, since you know the period, you also know the absolute luminosity. Since you can measure the apparent luminosity, you can use the absolute and apparent luminosities to solve for the distance. There is some error band involved, so most of these calculations are done with several Cepheid variables found in a single large structure, like a galaxy - so that you can get a statistically meaningful sample.

ADDED: Yes that's about right. Apparent mag doesn't help all by itself. If it did, we would know the distance to all the stars simply by measuring their magnitude. But if you know the absolute magnitude - the luminosity of the star when you are right next to it - undimmed by distance - and you also know the apparent mag as measured in your scope - then you can determine its approximate distance by the inverse square law.

The interesting thing about Cepheid Variables is that their period of fluctuation varies in a predictable fashion with their absolute magnitude, or luminosity. Once we measure the period, we can deduce the absolute luminosity. Once we know the absolute luminosity, we can measure the apparent luminosity, thereby giving us the tools needed to determine the distance. I do not know the relation between the absolute luminosity and the period, so I don't know whether it is based on the average or the peak, or both. But that's the approach.

The period luminosity relation can be used to estimate distance as the period measurement with standard candle gives idistances

The period of a Cepheid is related to it's luminosity. If you know that you know it's distance.