There are a great number of schools of thought on how the universe will ultimately "meet its maker" (if you will excuse the euphemism). Here are a few of the more significant ones.

The most classical and often accepted is that it will ultimately experience "heat death." This may be the case if we are in an expanding universe or in a universe where, in what would be basically an infinite amount of time, the Universe would stop expanding. According to the second law of thermodynamics, heat cannot flow from a cold object into a hot one without causing more entropy. Someday, all of the energy will be used up, the universe will be plunged into complete and utter chaos, and therefore there will be no stored up energy left in the entire universe. That means that nobody can do any work (a good thing). It also means that all movement everywhere will cease (a bad thing).

It will never end. Ever. To elude the first and third possibilities, the universe would have to have an infinite amount of reverse entropy stuffs laying around.

The third is the most interesting to me: if the Universe has enough mass to stop from the Big Bang, it will then contract. This means that a kind of reverse entropy will occur: things will go from chaos into order. Broken things will reform! Time will reverse! Cats and dogs living together, the universe as we know it will end!

(slightly updated, embarrassingly soon, due to things pointed out by ariels. Thanks for setting me straight))