Cepheid Variables are a key class of variable stars used to determine distances to galaxies and star clusters in the universe, because their periods are directly related to their luminosity. If you measure the period of their optical fading and brightening, this gives you the luminosity of the star, and from this you can get the distance to it.
Eclipsing Variables are pairs of stars orbiting each other in a plane that is almost edge-on to the observer. The stars eclipse one another periodically. By knowing their distance, you can determine the scale of the system and directly weigh the stars. This is our only method of determining just how massive a star is.
RR Lyrae variables are more numerous than Cepheids, but are intrinsically less luminous, and vary over time less than a day or so. Their luminosities are pretty constant, so their periods do not tell you their luminosities. They are bright enough that they can be used as 'standard candles' to determine distances to objects within our Milky Way and neighboring galaxies, so they are an independent check on the Cepheid distance method.
There are lots of other kinds of variable stars, too. These are important in understanding the various evolutionary phases of stars of different masses and ages.