Cepheids brighten and fade on a clock. Delta Cephei is the namesake. The useful part is the clock: the period tells you how bright the star really is. That is how astronomers measure distance.
Pulsation
They expand and contract. Expand: brighter. Contract: fainter. The cycle can run from a day to months.
Henrietta Swan Leavitt found that longer periods mean more luminous stars. Measure the period, you get the true brightness. Compare that to how bright it looks, and the inverse-square law gives distance.
Classical Cepheids are young, massive, usually in spiral arms. Type II Cepheids are older, fainter, often in globular clusters.
Why anyone cares
They are standard candles. Hubble used them to put Andromeda outside the Milky Way. Before that, it could have been a nebula in our own galaxy. After that, it was another island of stars.
Cepheid distances also calibrate other rungs on the distance ladder: Tully-Fisher for spirals, surface-brightness fluctuations for ellipticals.
The engine is helium in the outer layers. Compress, heat, ionize, opacity rises, heat traps, the star expands. Cool, recombine, light escapes, it falls back. Repeat.
From here
The period-luminosity trick is why Cepheids matter. The helium cycle is why they pulse. That is enough.
From Georgetown's light-polluted skies we may not resolve distant Cepheids without equipment, but the same physics is sitting over the seawall.