The Sun will blow off its outer layers and will exist as nothing more than an exposed, dead core. This is known as a white dwarf. It'll be mostly carbon and oxygen, and will have a radius of only about 1/100th of the Sun in its present state. It will continue to shine for a very long time, since it'll be hot and have a small surface area. Eventually, it'll cool to the point where it will no longer emit any visible radiation. It will become a "black dwarf."
There's a 7 minute video which is informative (this is a bit tricky to d/l. Right click on "The Future of Our Sun" and 'save target as'. In the 'file name' box, replace the second suffix .avi with .bz2 so the file name reads sun.avi.bz2, then in the 'save as type' box, change to 'all files', then save the file to your computer. Once saved, the zipped file can be opened with bzip2 or winRAR software. Alternatively, the video is posted on YouTube under the same title).'