In their later stages of evolution as red giants or supergiants, stars often become variables -- their light output changes with time. These stars change in both size and temperature which causes a change in brightness:

In these variable stars, the core will increase its energy output causing the outer layers to heat up and expand. The outer layers over-react, expand too far, and begin to cool off. The outer layers contract because they have cooled off. The contraction causes the core to heat up again and increase its energy output which causes the whole cycle to start over. Pulsing red giant from http://www.lcse.umn.edu/MOVIES/ (reload to restart animations)

The star alternates between expanding and contracting, between heating and cooling. Finally, its core cools out of sight -- it is about to collapse into a white dwarf/planetary nebula (to be discussed soon).