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.
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.