The sun has been our star for 5 billion years, and will continue to shine for another 5 billion years. Nothing to worry about. Once all the hydrogen in the sun has been converted to helium, our star will swell up to reach a size close to the Earth's orbit. All life on our planet will cease at that point. Once all the helium has been converted to the next nuclear fusion element, beryllium (element #4), the sun will shrink and cool down to a white dwarf star, barely the brightness of the full moon - as seen from what's left of the Earth. In the end, the sun will go out, leaving a cinder - a black dwarf.
Stars that are much heavier than the sun will end their life as a black hole, which carry a gravitational pull that exceeds the speed of light. Nothing can escape from their grasp, and there are theories around that state that even time gets warped within a black hole. Nobody knows for sure.
Professor Brian Cox, whose programme prompted this post, states that it is the nature of the universe to progress from order to chaos, a state of decreasing entropy. He referred to that as the Second Law of Thermodynamics. By virtue of that, all the remnants of dead stars, whether they be black holes or black dwarves or whatever, will decay like a sandcastle on a windswept beach - to nothing. In the end, all matter will revert to energy (as per Einstein's E=mc2), and the universe will cool to absolute zero.
That, however, is where I start to disagree. Apart from the 2nd law of thermodynamics, the universe is governed by a few more laws. Gravity is one of them. And it is quite feasible, in my mind, that all matter will be pulled together by gravity into one entity - like the universe began? There is another law at play. The law of conservation of energy, the 1st law of thermodynamics. If the universe cools down to zero, that means that it will lose all the energy that is around at the moment. The 10 to the power of 77 atoms that slosh around the universe on a daily basis represent an ever large number in terms of energy. Energy is shown in motion, light, heat, radiation - and although it can be diluted, it cannot be lost.
I am not a physicist or astronomer, although I know a little bit about both. But isn't this fascinating!