We all know now that the Universe is expanding, and at an accelerating rate. What happens, though, if it expands too far? Well, it turns out that if the Universe's density exceeds its "critical density", all of its matter will be mutually gravitationally attracted to each other, causing the expansion to cease and then reverse. This phenomenon, that all matter will eventually collapse in on itself into a black hole singularity, is known as the Big Crunch.
What happens, though, after all matter has dissolved into this black hole? It's possible that another Big Bang could occur, thus indicating an infinite cycle of Big Bangs and Big Crunches, meaning the Universe lasts forever in this fashion.
There are obviously scientific debates surrounding this theory. One camp believes the Universe will forever expand, and the other believes in this Big Crunch idea.
So what if the Universe's density is less than the critical density? Well, it will keep on expanding until its matter becomes thinner and thinner. By this point, the Universe will become essentially lifeless.
Finally, if its density is equal to the critical density, the Universe will expand to an equilibrium point and then come to a halt. There are obviously several other theories, some of which involve the role of dark energy and matter that stimulates cosmic acceleration and leads to creations of other separate universes.