All stars, even our own Sun, are slowly dying: their light comes from the fusion of hydrogen atoms into helium into their core, and eventually, all stars simply run out of hydrogen. Stars that are roughly those the size of our own Sun will expand enormously just as they start to run out of that hydrogen fuel, and at the same time begin to turn red as their surface temperature cools. The resulting star is known as a "red giant."
For the vast majority of its normal life, a star functions as a relatively stable thermonuclear reactor, fusing hydrogen into helium in its core. However, no matter the size of the star, there is a limit to how much hydrogen is available to undergo the fusion process, and the star will eventually burn out, usually spectacularly - either by expanding enormously into a red giant, or, for stars much bigger than our own Sun, exploding as a supernova. Much more serious to human beings, in the process the Sun will also become incapable of sustaining life on Earth, although this will not happen for several billion years.
For stars the size of our Sun, the process is fairly predictable. There is enough hydrogen present in the star's core to sustain the hydrogen fusion process for about ten billion years. After that, the star will, for just a few million years, expand rapidly, becoming a red giant. Then, gradually, it will shrink again, ending its life as a glowing remnant known as a white dwarf.
BECOMING A RED GIANT
When the hydrogen in the core is used up, the fusion reaction there stops for the first time in billions of years. This does not mean the star dies immediately: instead, a similar process begins to happen in the much larger envelope over the core, known as the "shell" of the star. The shell occupies a much larger volume than the core itself; as a result, hydrogen fusion suddenly begins to occur over a much wider area of space, and, to an observer, the star very quickly becomes as much as a thousand times brighter.
At the same time, the star begins to grow rapidly as the atoms in the shell heat up, fuse, and expand. In our own solar system, for example, the Sun will grow so large that it absorbs Earth's current orbit. Mercury and Venus will certainly fall into the Sun and be destroyed in the process. Whether Earth will as well, or will be pushed away into a more distant orbit, is currently a subject of debate, with the majority of scientific evidence currently leaning in favour of the first of these due to tidal interactions. In either case, it would certainly become far too hot on Earth for life to survive.
At the same time as this expansion is occurring, the surface of the star will actually cool - as much heat is being produced as before, but it is now being radiated out over a much larger surface, so that the surface as a whole experiences a fall in temperature. To an observer, it will begin to darken, changing from yellow to red - hence the name, "red giant."
The red giant phase of the star's life is not its death, but rather its last gasp of life. The hydrogen supply in the shell will last only a short while, perhaps a few million years - a long span to human observers, obviously, but much shorter than the original ten-billion-year lifespan of the star. The end of this phase is marked by a brief fusion of helium into carbon, after which the star begins to shrink rapidly. The outer layers are shed outwards to form the beautiful clouds of gas and debris we recognize in our telescopes as nebulae. The remnant of the core will still glow with stored-up energy, probably for millions or even billions of years, and is referred to as a "white dwarf."