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).
Everything is measured with atoms (upon atomic properties) 

oooooo < these are atoms in the lab now  

|\_\_\_\_\_| This space at left is $\approx 5\ $ units long . 

**OOOOOO** < the very same atoms in the past 
 radiated at longer wavelengths (redshifted).  

|\_\_\_\_\_| This space at left is $\approx 3.5\ $ units long. 

Is it clear that any space expansion is an artifact of the measuring process ? 

If you do not beleive in your eyes and intuition please present one fundamental equation (from all physics) able to discriminante between the atoms version 1 and version 2 (besides the radiated light wavelengths as said above). 

For those that want to check the validity of, or deny, my viewpoint please comment the proof in this document "[A self-similar model of the Universe unveils the nature of dark energy](http://vixra.org/pdf/1107.0016v1.pdf)". 

In physics there is no mention to any *absolute or invariant* size of the particles. This absolute notion was never questioned before and is an hidden postulate of the standard Model. 

Because the "oos/OOs" is not clear enough here is the balloon analogy: 

[![matter shrinking](https://i.stack.imgur.com/y5w0n.jpg)](https://i.stack.imgur.com/y5w0n.jpg)

Explaining "more field" in ligth blue in the 'present' balloon:  

The light cone of the fields associated with the particles grows as time goes by. Thus, the overall energetic content of particles+field is constant thru time.  

The next ideas are not part of the Model because it is speculation:  

There are infinite events of matter creation, with CMB, in a cyclic way.  

The one versioned Universe of Standard Model, with a begining and an end, is phylosophically repugnant to my mind. I'm a rationalist and I fully respect the PSR-Principle of Sufficient Reason as Spinoza, ...Poincaré, and Einstein did.