You may think you know how to ride a bicycle, but the chances are you don’t know how to turn a corner. Wilbur Wright (yes, that one) said:
I have asked dozens of bicycle riders how they turn to the left. I have never found a single person who stated all the facts correctly when first asked. They almost invariably said that to turn to the left, they turned the handlebar to the left and as a result made a turn to the left. But on further questioning them, some would agree that they first turned the handlebar a little to the right, and then as the machine inclined to the left, they turned the handlebar to the left and as a result made the circle, inclining inward.
He’s talking about countersteering. To turn left, you have to lean to the left, and to do this, the handlebars must first turn to the right. There is no other way to turn left. And yet most people I’ve told this simply don’t believe me.
Obviously you have to lean to the left to turn left, otherwise the bike will fall over. Most people just think about leaning the bike over and aren’t conscious of what they’re doing with their hands. No real force is required on the bars; just lean your body left and the steering will turn enough to the right on its own. But if you do consciously countersteer by pushing on the left handlebar, you discover that the bike is much more maneuverable than before.
On motorcycles this is an important safety technique. A common cause of accidents is riders not making it round a corner and crashing into something on the outside of the corner. Often the rider thinks he is going too fast and brakes. Braking makes the bike stand up straight, which makes it go in a straight line – crash! The correct thing to do when you find yourself going scarily fast round a left hand bend, is to give the left bar a good shove. The bike will lean over more and round you will go.
Because it is so counter-intuitive, I think it is important to consciously countersteer all the time, so as to program your brain with the correct action when there isn’t time to think about it.
The wikipedia page has a full explanation of what’s going on and this video has quite a good demonstration.