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: 
After the Big Bang, the universe was left already expanding. Imagine sitting in a car. You step on the gas and get it to top speed. Then you put it in neutral and turn off the engine. What happens? You keep moving forward. But why? The car can't take off by itself. Friction should prevent it from moving. This answer is obvious, the car keeps moving because it has inertia; momentum. Similarly, the universe kept expanding initially; back when radiation and matter dominated the universe. In those eras, gravity, as you'd expect, caused expansion to slow. Much like friction and air resistance slows your car. Then, when dark energy became dominant in the universe, its negative pressure caused the universe to begin accelerated expansion. Dark energy is like negative friction. I know that doesn't make sense, but you have to expect that not every crazy and wonderful thing in the universe has a simple analogy to things in everyday life.

I see in John Rennie's answer, he described dark energy as a sort of anti-gravity. Here, I describe it as a form of energy itself that counteracts the gravitational influence of normal matter and radiation. You might think those are two different things and that only one of us can be right, but no! Dark energy can easily be described as either a part of gravity or as a form of energy. It's completely up to you. Both are valid and each just represents where you want to include dark energy in the gravity vs. matter-energy equation.

But Jim, you absent-minded artichoke, you forgot to mention why the universe had such a large initial expansion in the first place. Oops, you're right. We can't give you a single good, accepted reason for the initial expansion of the Big Bang. But we can tell you that very soon after the initial curvature singularity (that's the Big Bang), if it existed, inflation drove expansion. This was the part where you stepped on the gas. Inflatons, which are quasi-particle things present during inflation, caused the universe to expand almost exponentially. If you don't like comparing it to stepping on the gas, then it's like if the car started on a large hill. In park, nothing happens. But put the car in neutral and it'll just start rolling down the hill and then continue once it hits the level area at the bottom. Still with me?

Now let me be clear, in my analogy, the car doesn't represent any galaxies or matter or nonsense like that (us cosmologists don't really care about anything *in* the universe), it represents the expansion of space itself. Because that's what's expanding; space. The speed of the car represents the rate of expansion; basically the amount of distance that's added per second between two distinct points some initial distance apart. So when I say dark energy drives acceleration of expansion, that's like stepping on the gas. The car accelerates, representing an increase in the rate of space expansion.

Now some of you readers out there may be thinking "But Jim, you foggy London morning, you're still unclear". That may be true (the unclear part. I resent being called a morning), and if something is still bothering anyone about this, I can point you to some excellent other posts on the site that speak about the expansion of space or any other weird and wonderful cosmological ideas. But for now, I think the question is sufficiently answered, so I'm going to end there.
Ok I am no professional physicist, so feel free to scrap this if you want to, but as I have had explained to me: the expansion of space is the phenomenon that distances increase, *without* relating to any motion. 

It is the space in between atoms and galaxies that gets "stretched", not that objects "move" away from each other. Like you blow a ballon up, points on the balloon get further away, but it is because of the fabric of reality itself that is expanding and not any motion of the matter in it.