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Energy Output
Like most stars of its kind, the Sun is composed mainly of Hydrogen and Helium (for our Sun, the composition is about 90% Hydrogen, 8% Helium, and 2% trace elements). Every second, the Sun is converting hundreds of millions of tons of Hydrogen into Helium through nuclear fusion, a process that produces significant quantities of energy by colliding two atomic nuclei together and fusing them into a new atomic nucleus. However, unfortunately for us, there are other, more powerful nuclear fusions going on at the center of the Sun: the conversion of Helium into larger molecules, such as Carbon. The simultaneity of these two processes will eventually become a problem for us: as more Hydrogen is converted into more Helium, more Helium can be converted into larger molecules, and the heat of the Sun will continually increase.1
Scientists estimate that in approximately four billion years, the temperatures on Earth will be similar to those on Venus! Our atmosphere will be superheated and dissipate into space. Water will not be able to exist in liquid form on Earth. Our bodies of course will not withstand this much heat. So unless we find a new "solar system" to occupy, we will be toast.2 Literally.
The Red Giant Phase
There is another consequence to the increased energy output of the Sun as it goes through its life. The Sun is composed of multiple layers, generally classified under the broad terms of "the core" (the center) and "the envelope" (the outer layers). Since the envelope surrounds the core, all energy produced at the center of the Sun travels outward and has the ability to heat up the outer layers. As the Sun starts to produce more and more energy, the heating up of the outer layers will cause them to expand (the expansion of materials as they heat up is given by many principles, most notably Charles' Law; consider a hot air balloon, which utilizes heat to expand the air inside the balloon and allow it to rise). When the Sun finally stops expanding, it will be in its Red Giant phase, where it will remain for approximately 100 million years.3
Scientists estimate that, when the Sun stops expanding and reaches its Red Giant state, it will be over 200 times the size it is today. This means that the Sun will have essentially consumed Mercury, Venus, and our Earth (many think it will also cover the orbit of Mars).4 It should be pretty clear why this is a problem. If we are not already vaporized by the intense heat, we will be physically destroyed by the body of the Sun.
Burning Out
As the Sun enters its Red Giant phase, the outer layers will slowly lose their gravitational attraction to the core. This will cause two important processes to occur. First, no longer constrained by gravity, many layers in the envelope will be lost into free space. Thus, over time, the Sun will "shrink", though it will really just go through a process of losing matter until only the gaseous core is left. Second, the lack of gravitational compression in the core will cause the process of nuclear fusion to stop, since there will no longer be any force capable of fusing atomic nuclei together. So, the Sun will be left as a small (Earth-sized) ball of inert gas called a White Dwarf: it will essentially be dead.5 So, even if Earth somehow manages to survive extreme heat and physical collision with the Sun, life on Earth will no longer be possible. Without nuclear fusion in the Sun's core, the Sun essentially becomes useless to us. We will no longer have a source of energy to power our planet.
Every object in the universe must meet its end, and it just happens that the fate of the Sun and the fate of life on Earth are inexorably tied together. I guess we can only dream of the day when we find a way to escape from our Sun and travel to another system powered by a fresh, young star.
Sources:
1. http://hea-www.harvard.edu/scied/SUN/sunpage.html
2. http://www.universetoday.com/18847/life-of-the-sun/
3. http://www.astro.cornell.edu/academics/courses/astro201/evol_sun.htm
4. http://www.universetoday.com/12648/will-earth-survive-when-the-sun-becomes-a-red-giant/
5. http://imagine.gsfc.nasa.gov/docs/science/know_l2/dwarfs.html
yikes!