When an average-sized star, such as the Sun, runs out of hydrogen in its core, the remaining hydrogen in it's outer layers expands, and the star becomes a red giant. At this point, our sun will reach about 100 times its current size.
When the core becomes hot enough for the fusion of helium to begin, an explosion occurs called a helium flash and the star rapidly shrinks to a fraction of its original size, becoming a white dwarf as the outer shells run out of hydrogen. Nuclear fusion continues until it the remainder of its helium is depleted, as the white dwarf gradually becomes a black dwarf.
When a white dwarf cools to a point such that it no longer emits significant heat or light, it is considered a black dwarf. Essentially a dead star, this is the last step in the life of an average-sized star.