A Cepheid Variable (often shortened to Cepheid) is a type of Variable Star that varies in Temperature and Luminosity with in a regular cycle. They are named after the first such star identified, Delta Cephei. There is a strong relationship between their period and brightness which allows them to be used as a Standard Candle as part of the Cosmic Distance Ladder.
There are two populations, Classical Cepheids (aka Type I Cepheids) or Type II Cepheids, each with its own period/brightness relation, so which of these populations it falls in must be determined to use it to determine distance. They can be observed in nearby galaxies, allowing determination of distances for galaxies not providing good data by Type Ia Supernovae or Redshifts. Periods vary from a day to fifty or more.
Classical Cepheid Variables are yellow giants. Their period/brightness relation is:
Mv = (-2.43 ± 0.12)(log10(P)-1) - (4.05 ± -0.02)
Cosmic Distance Ladder
Stellar Distance Determination