Cepheid variables are a type of star which pulsate inward and outward changing in size and temperature at a stable rate.
The first such star was discovered in 1784 by astronomer John Goodricke. The star was called Delta Cephei since it is located in the constellation Cepheus. In honor of this star, all stars of the same type are called Cepheids.
Cepheid variables are divided into several classes: Classical Cepheids, type II Cepheids, and anomalous Cepheids.
Classical Cepheids
Also known as type I Cepheids, are stars which undergo pulsations with very regular pulsations in the range of days to months. They are 4-20 times more massive than our Sun and up to 100,000 times more luminous. Their radii change by millions of kilometers as they pulsate.
Classical Cepheids are used to measure distances to galaxies within the Local Group. and are a means for determining the Hubble Constant. They have also helped to clarify many characteristics of our galaxy. such as the Sun's height above the galactic plane, and galaxy's local spiral structure.
Type II Cepheids
These stars typically pulsate with periods between 1 and 50 days. These are typically low mass objects, at about half the mass of our Sun.
They are further divided into several subgroups based on their period. BL Her subclass pulsate with a period between 1 and 4 days. W Virginis subclass have a period between 10-20 days. RV Tauri subclass have a period greater than 20 days.
These type of Cepheids are used to establish the distance to our Galactic Center, to globular clusters, and galaxies.
Anomalous Cepheids have periods of less than 2 days and more massive than our Sun.
Other important facts
Cepheids were used to put initial constraints on the size of the Milky Way.
In 1942, Edwin Hubble established distance to Cepheid variables in what was than known as the Andromeda Nebula. He showed that these stars were not part of our Milky Way galaxy, and thus that our galaxy is not all of the universe, but rather one of many galaxies in the universe.
There are inherent uncertainties pertaining to attempts at measuring distances to galaxies using Cepheids. These unresolved issues have led to variation in the value given to the Hubble Constant, which in turn could effect the established value of the size of the universe.
Why do they pulsate?
The current accepted explanation is called the Eddington valve. Helium is thought to be the major player in the process. As Helium is heated, it becomes ionized. The more it's heated, the more ionized it becomes. After some time of heating, it will loose one of it's electrons, which is what ionization means. If it gets heated even more, it will loose both of its electrons. Helium with both its electrons missing looks more opaque than helium with just one electron missing. When the Cepheid is at its dimmest point, the ionized gas in the outer layers of the star is heated by star's radiation. Due to increased temperature, it begins to expand. As it expands, it becomes less ionized, and more transparent, allowing the radiation to escape. Then the expansion stops and the gas is pulled back in by star's gravity.
Sources:
https://en.wikipedia.org/wiki/Cepheid_variable
Image Source: http://starchild.gsfc.nasa.gov/docs/StarChild/questions/cepheids.html