The Big Rip theory is basically an idea that dark energy (a repulsive force that pervades all of space and is pushing the Universe apart at an ever-increasing rate) has some sort of feedback mechanism that causes it to increase in density over time. This would eventually counteract other attractive forces (like gravity and the electromagnetic force) at smaller and smaller scales, pushing apart first galaxies, then star systems (causing planets to float away from stars), then stars and planets, then small objects like people (we would explode sometime in the last second or so of the Universe's normal existence), and finally atoms and subatomic particles. After this point, the density of dark energy would be so high (possibly infinite, depending on the model) that physics would no longer work in the ways we're used to, and the Universe would become a very strange place where our normal understanding of geometry, topology and perhaps even causality would no cease to be meaningful.
Keep in mind that this is only one idea. The leading theory for what will happen as the Universe ages is the model of the Heat Death, which simply involves the Universe getting larger and larger and everything spreading out and becoming thin and cold until finally only photons are left. This is still a rather frightening proposal for the fate of the Universe and intelligent life, but perhaps not quite so frightening- nor so immediate- as the Big Rip theory.
You can read more about these here:
http://en.wikipedia.org/wiki/Big_Rip
http://en.wikipedia.org/wiki/Heat_death_of_the_Universe
http://en.wikipedia.org/wiki/Ultimate_fate_of_the_Universe