Question: We live in an expanding universe - so I'm told. But how can that be possible? Everything imaginable is attracted by a bigger thing. So, why can't gravitation stop the expansion of the universe? I know the "Big Bang" theory, but is it possible that the expansion of the universe is caused by the attraction of a bigger object?

Answer: 
Suppose you throw a ball up into the air. You could ask how the ball manages to move upwards when gravity is pulling it down, and the answer is that it started with an upwards velocity. Gravity pulls on the ball and slows it down so it will eventually reach a maximum height and fall back, but the ball manages to move upwards against gravity because of its initial velocity.

Basically the same is true of the expansion of the universe. A moment after the Big Bang everything in the universe was expanding away from everything else with an extremely high velocity. In fact if we extrapolate back to time zero those velocities become infinite. In the several billion years following the Big Bang gravity was slowing the expansion, in basically the same way gravity slows the ball you threw upwards, but the gravity didn't stop the expansion - it only slowed it.

The obvious next question is how did the universe get to start off expanding with such high velocities, and the answer is that we don't know because we have no theory telling us what happened at the Big Bang.

There is a slight complication that I'll mention in case anyone is interested: dark energy acts as a sort of anti-gravity and makes the expansion faster not slower. This has only become an important effect in the last few billion years, but as a result of dark energy right now gravity isn't slowing the expansion at all - in fact it's making the expansion faster.
update: the accepted answer has now been updated to make my answer superfluous.

**we totally don't know.**

we don't know why the universe started expanding at the beginning of time. we basically just shrug and say "it seems like there was this big bang". your question seems to imply that gravity should have stopped the expansion by now, which makes sense. originally, we thought that the universe was still expanding simply because **it hasn't been long enough since the big bang for gravity to stop the expansion**. but now we know, the universe appears to be expanding faster over time, rather than more slowly. we don't know why that is happening either. ask a physicist why the universe is expanding and he will say "dark energy". then ask him what dark energy is, and he'll say "the thing that makes the universe expand". this circular definition reflects the fact that we dont' really know why, but **we expect to find an explanation that fits our mathematical models. in the mean time, we call that something "dark energy"**.

i don't feel qualified to address the math or physics of your question about a "bigger object" pulling the universe apart. but from a philosophical perspective, it seems to be an unproductive [homunculus argument](https://en.wikipedia.org/wiki/Homunculus_argument). although, i admit i like the parallels it draws between [vacuum polorization](https://en.wikipedia.org/wiki/Vacuum_polarization) and the big bang.

side note: John Rennie's answer is also excellent (as usual). i felt he kind of dodged the question by illustrating what we do know (or theorize), but he has since updated his answer to include the points about which we are ignorant (dark energy and the big bang). we just don't have a good explanation for some things yet, and it is good to admit that. to quote einstein: [the larger the circle of light, the larger the perimeter of darkness around it](http://www.goodreads.com/quotes/143906-as-our-circle-of-knowledge-expands-so-does-the-circumference).