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: 
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).
The attraction of any two objects is proportional to the product of their masses times the distance between them squared. This force always accelerates the objects toward each other. But that "distance between them squared" term means that the force drops off faster than the velocity does. This means that any two objects that are initially moving away from each other will be continuously slowing down, and there can be two outcomes:

1) They eventually slow down to the point where they start moving back toward each other, which happens if their initial relative velocities are below a certain threshold for a given pair of objects, or
2) Above this threshold, the gravitational force drops off faster than the velocity does, so that while they continuously decelerate, they never get down to zero relative to each other. Remember, gravitational force drops off with the square of the distance.

Based on the estimated total mass of the universe and the observed speed at which everything is moving away from each other, it is believed that the universe is in the latter condition, and will always be expanding.

Or to put it another way, on average, everything in the universe is moving away from each other faster than their mutual escape velocity.