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).
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> How can the universe expand if there is gravitation?
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Because gravity alters the motion of light and matter through space. But *it doesn't make space fall down*. 

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> We live in an expanding universe - so I'm told. But how can that be possible?
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Because the expanding universe is something like a stress ball. Squeeze it down in your fist, then let go. It expands. 

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> Everything imaginable is attracted by a bigger thing. So, why can't gravitation stop the expansion of the universe? 
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Because a gravitational field is a place where space is "neither homogeneous nor isotropic", this being modelled as curved spacetime. See the [Einstein digital papers](http://einsteinpapers.press.princeton.edu/vol7-trans/192?highlightText=%22neither%20homogeneous%22): 

[![enter image description here](https://i.stack.imgur.com/gIswA.jpg)](https://i.stack.imgur.com/gIswA.jpg)

And on the very largest scale space is homogeneous and isotropic, see the [FLRW metric](https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#General_metric). There is no overall gravitational field in the universe. And even if there was, it wouldn't stop space expanding. 

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> 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?
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No. A massive body such as a star "conditions" the surrounding space. It *alters* it, and this effect diminishes with distance. As a result there's a gravitational field. But the space doesn't fall down towards the star. The [waterfall analogy](http://jila.colorado.edu/~ajsh/insidebh/waterfall.html) is badly misleading in this respect. In similar vein expanding space is not falling up towards some other object. NB: dark energy is not gravity: gravity is *not* making the universe expand faster.