When hydrogen fuel at the centre of a star is exhausted, a shell of nuclear reactions will start to move outwards into its atmosphere. As a result, the outside of the star starts to expand and cool, turning much redder. Over time, the star will change into a red giant and grow to more than 400 times its original size.
As they expand, red giants engulf some of their close-orbiting planets. In the Sun's case, this will mean the fiery end of all the inner planets of our Solar System, including the Earth; but don't worry, this won't happen for another 5,000,000,000 years.
While the atmosphere of the star grows, its core shrinks due to gravity. Temperatures and pressures in the middle increase until the conditions are right for nuclear fusion to start again, but this time using helium as a fuel, rather than hydrogen.
With the star being powered by hellium, its outer layers return to normal for a while and it starts to shrink, get hotter and turn a little more blue. However, this stage only lasts for a million years or so, as the helium quickly runs out. When it does, the star starts to grow, cool and turn red again as it enter its second red giant phase.
What happens next depends on the mass of the star. Small sun-like stars move into a planetary nebula phase, whilst stars greater than about 8 times the mass of the Sun are likely to end their days as a supernova.