Ok, there are 4 possible ways the universe will meet its end. First, you have to understand that the universe is expanding away from a common center-point. What is expanding is not just the distribution of matter, but space-time itself. Think of it like a bunch of rocks (representing matter: galaxies, clusters of galaxies, superclusters, etc) sitting on a crumpled up piece of cloth (representing space-time). Now, pull all the corners of that cloth outward, slowly, at the same time. The rocks will get further apart, but it was the cloth (space-time) that moved.
Now that you have that, understand that recently astronomers discovered that the rate of acceleration of this expansion is increasing. This was a big surprise as it was supposed that gravity, which should pull everything toward that center point, would cause the acceleration to slow down. Scientists now suppose there is a greater energy that is stronger than gravity that is speeding the acceleration up (called dark energy atm).
Ok, now that you understand all that, I can explain the 4 possible deaths of the universe:
- The Big Rip: The rate of acceleration of space-time keeps increasing: In this case, the universe keeps expanding faster and faster and faster...dark energy just keeps pulling it. At present gravity (which is relatively weak, just really widespread) is overcome in this manner, but if it keeps going faster, then eventually the (relatively) stronger physical forces: electromagnetism, the strong nuclear force and the weak nuclear force, will all be overcome. Basically, even subatomic particles would be pulled apart because the universe is expanding too quickly.
- The Big Chill: The rate of acceleration stabilizes, but the universe keeps expanding: In this scenario, the universe's expansion speed stabilizes (or at least doesn't hit the critical threshold as in the Big Rip), but the universe does keep expanding. Eventually, the universe would become so vast, and then combined with the laws of thermodynamics making matter equalize its distribution in the cosmos, that matter would become very scarce and extremely spread out. There would be a single photon (light particle) for every many millions of light years. The universe will then become an extremely cold place.
- Death by Black Hole: The universe stops expanding, and does not contract: Here the universe stabilizes in size, and does not meet the threshold for the big rip or the great crunch as below. Basically, the universe would just stabilize. The end here is that gravity would ultimately win, and everything would eventually be caught in a black hole.
- The Great Crunch: The universe stops expanding, and contracts back to its center point. Basically, all matter would become so hot (and small) that the universe and its laws would be unrecognizable from what they are now. After the crunch, there may be another big bang to start it all over again (which is the oscillating universe) or there may not.