Solar System -
The Sun formed, as we have said about 4.57 billion years ago, when a cloud of relatively cool, tenuous gas started to collapse. This may well have been caused by a nearby supernova. As it contracted, the internal temperature began to rise until at around 5 to 10 million°K, nuclear reactions started to convert hydrogen into helium in the proton-proton cycle. There is another cycle, the Carbon/Nitrogen/Oxygen cycle, but the Sun is not large enough to produce the temperatures necessary for this to be significant. The energy trying to escape counteracts the force of gravity so the Sun stopped contracting and reached a level of stability it retains to this day.
As the Sun burns hydrogen, helium builds up in the core, and the Sun's temperature rises. In about a billion years, it will have grown about 3% larger than it is today, and will be about 10% brighter. The temperature on the Earth will rise substantially; possibly to the point that the seas and oceans start to boil off. At this point, the Earth would no longer support life. The Sun will continue to grow and become more luminous as more hydrogen is converted to helium. In about 5 billion years, once all the hydrogen in the core is converted to helium, the core stops producing enough energy to fight gravity, and starts to contract, causing the core to heat up. The heat is sufficient to start hydrogen burning in a shell around the core. This causes the outer layers to expand further enlarging the sun. This is the start of the Sun becoming a red giant.
Once the temperature in the core reaches 100 million°K, another nuclear reaction starts where helium is converted into carbon via the triple-alpha process where two helium-4 atoms combine to form beryllium-8, which then combines with a third helium-4 atom to produce carbon-12. In stars like the Sun, helium burning starts very quickly due to the presence of degenerate matter in the central core from the extreme pressure exerted during contraction. This is known as the "helium flash". After this, the sun settles down to stable helium burning that should last for about a billion years. The heat produced causes even more expansion of the outer layers, and ultimately the sun will grow to about 200 to 250 times its present diameter, even though it looses perhaps a third of its present mass due to the solar wind produced. This will make it so large that it will extend beyond the Earth's current orbit, though the Earth will likely have moved further out due to pressure from the solar wind. In any event, the surface of the Earth would be extremely hot, and it will have lost all its water and most of its atmosphere.
Once all the helium is converted to carbon, the core again contracts. The Sun is not large enough to be able to produce the temperatures required for any further fusion. Thermal disequilibrium will cause it to blow off its outer layers to form a planetary nebula. Here is an example of what the planetary nebula could look like. The remaining core contracts until electron degeneracy pressure stops any further contraction. What is left is an extremely hot white dwarf star about the size of the Earth, but weighing about two thirds the current mass of the sun. With no nuclear reactions occurring, the white dwarf slowly radiates its heat away, cooling down and becoming redder and redder. Hypothetically, white dwarfs eventually give up all their heat to become black dwarfs, and in the Sun's case, theory indicates that this is likely to take at least 500 billion years, perhaps even more.
Complete life cycle of the sun, from birth to white dwarf. Click for an enlarged version.