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: 
Suppose you throw a ball up into the air. You could ask how the ball manages to move upwards when gravity is pulling it down, and the answer is that it started with an upwards velocity. Gravity pulls on the ball and slows it down so it will eventually reach a maximum height and fall back, but the ball manages to move upwards against gravity because of its initial velocity.

Basically the same is true of the expansion of the universe. A moment after the Big Bang everything in the universe was expanding away from everything else with an extremely high velocity. In fact if we extrapolate back to time zero those velocities become infinite. In the several billion years following the Big Bang gravity was slowing the expansion, in basically the same way gravity slows the ball you threw upwards, but the gravity didn't stop the expansion - it only slowed it.

The obvious next question is how did the universe get to start off expanding with such high velocities, and the answer is that we don't know because we have no theory telling us what happened at the Big Bang.

There is a slight complication that I'll mention in case anyone is interested: dark energy acts as a sort of anti-gravity and makes the expansion faster not slower. This has only become an important effect in the last few billion years, but as a result of dark energy right now gravity isn't slowing the expansion at all - in fact it's making the expansion faster.
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.