These are stars that vary with a regular period in their brightness. The first such star was recorded in 1784 by John Goodrick in the star delta-cepheus.
The magnitude of delta-cepheus varies between 3.6 and 4.3 in a period of 5.4 days. The light curve typically rises rapidly and then falls off more slowly.
The periods of these stars is from 1 day to about 100 days. Polaris is a cepheid star with a rather small magnitude variation of between 2.5 and 2.6 and a period of under 4 days. Most cepheids have absolute magnitudes in the range -1 to -5, and periods from 3 to 50 days. (Definition: Absolute magnitude is what the star would look like, that is the apparent visual magnitude, if we could place the star at 10 parsecs distance from us.)

Example from the HIPPARCOS catalog Hipparcos
Follow the link from "education" to "variable stars" on the Hipparcos home page.

(HD 107805, R Cru). This star is of spectral type
 F7Ib. It varies between about 6.483 and 7.318 magnitude  with a period of 5.8257 days. It is classified as a Delta Cephei-type pulsating variable. The errors on the measured magnitudes are invisible on this plot.  This is a plot for HIP60455.

An animation of the region where this star is located is
variable star

Henrietta Leavitt - original link is defunct - http://cannon.sfsu.edu/~gmarcy/cswa/history/leavitt.html
local link to Henrietta Leavitt
In the years between 1908 and 1912 she discovered a relationship between the apparent brightness and the period for 25 cepheid variable stars in the Small Magellanic Cloud.
Small Magellanic Cloud, NASA image

From her data she could infer that the absolute magnitude of these stars also was a function of the period, in particular,

brightness = k*log(period).

This determination could be made because all the stars she plotted were in the Small Magellanic Cloud. Even though she did not know the absolute distance to the SMC, she could still reasonably assume that the stars were all at the same approximate distance from us. The SMC is 7000 light years in diamter and 210K light years away.

Physical description

Cepheid variable stars are higher mass stars, mass>5 solar masses, in the class of yellow super giants ( F to G). The brightness variability of cepheids is intrinsic to the stars, and not caused by it being eclipsed by a binary partner.
The spectra from these stars reveal a great amount of information. The surface temperature at maximum light is greater than at minimum light. The changes in surface temperature are correlated with changes in the physical size of the star. This can be determined by the Doppler shift of the spectral lines during different phases of the light curve. The radius of delta-cepheus, for example, changes by about 7% during a period. The pulsating cepheid star can be likened to a mass on a spring. If the spring is stretched and released, the mass will shoot through the equilibrium point until it reaches the maximum compression, and then reverses direction. In the cepheid variable star the star expands when the force due to the internal pressure is greater than the weight of the star's outer layers. The star stops expanding once the pressure has fallen sufficiently, but now it is again away from equilibrium and collapses. The collapse goes through the equilibrium radius and the cycle begins again. Astrophysicists have determined that the period of oscillation is inversely proportional to the square root of the average density. The larger stars have lower average density.

There are several other categories of variable light stars besides the cepheid variable.

In 1923 Edwin Hubble found a cepheid variable while examining photographic plates taken at Mt. Wilson  .

This important find enabled him to determine the distance to the Andromeda Nebula, M31, and showed that it was outside of our galaxy. The great debate over the position of the nebulae was now over. They were indeed Island Universes, comparable in scale to our own Milky Way Galaxy. In fact, it is common today to argue that the galaxy is the fundamental "unit" of the Cosmos, rather than stars.

References

"Exploration of the Universe", George Abell, Holt Rinehar,Winston Inc., 1969

"Astronomy!: A Brief Edition", James B. Kaler, Addison-Wesley, 1996

"Modern Theories of the Universe from Herschel to Hubble", Michael J. Crowe, Dover Pub., 1994
http://www.mtwilson.edu/Education/History/cal89/cal1189.html