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: 
Something the other answers don't really delve into is this: **Space itself is what's expanding**

Let's think about a regular Newtonian Spacetime with zero expansion, borrowing Jim's example of a car on the road. You're sitting completely still, as is a friend you're following on a roadtrip, with about a hundred feet of distance between you. If I move towards them at a few feet per second, then my distance is going to decrease at the same rate. I'll also be approaching another car a few hundred feet ahead of them at the same speed, because again none of us are moving.

Now, if we were in a classical, flat, non-expanding spacetime that's more or less what we'd see. Galaxies would gravitationally attract, and eventually (Meaning over a literally infinite timeframe) everything would re-merge.

This is where things get bizzare, and well outside the scope of our daily experience. We're not on a road as we know it, **The road itself is growing. Space itself is expanding.**

Imagine that, somehow, every 10 feet of road is growing an extra foot every second. If I were standing still, I'd see my friend move away from me at about 10 ft/s for the first second. A car 500 feet away would move away at 50 ft/s, and a car a mile away would rocket off at **over 500 ft/s** If I started moving towards my friend at 20 ft/s, I'd only see him getting closer at a rate of 10 ft/s, and the car a mile away would still be speeding into the distance faster than I could catch up to it.

This is where the idea of the Observable Universe comes from. There are points in space that we can never see, because they're "expanding" away from us faster than the speed of light.
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).