Named for the first studied stars of these types. All low mass stars will become RR Lyrae or Cepheid variables stars during their life. So these star types are very common. Our sun will remain as a Giant star for a long time. But not nearly as long as it was a main-sequence star.
Larger stars are Giants for even shorter lengths of time.
The last stage may involve pulsations, variable-star behavior. But now the core fills with carbon, created from the fusion of helium. Fusion slows in the core, but energy production doesn't because the star makes up for less fusion by again contracting, releasing gravitational energy. Energy production in the core increases, pushing outward on the star's outer layers. This pulse of energy causes the outer layers of the star to be blown out into space.