Steering - why you go round corners
The angle of dangle as a component of the level of bullshit
The speed with which your bike turns is determined by several factors:

Steering head castor angle
Fork offset
Weight of front wheel / tyre
Length and weight of bike
Centre of gravity

A typical setup for a modern sports bike would be 24 degree head angle, 25 to 30 mm fork offset, 96 mm trail, 1400 mm wheelbase and around 175 kilos weight.

The figure below shows the relationship between the steering components. A quick study of this and it becomes pretty clear that altering any one of the steering settings will affect all the others. So, if the fork offset is reduced, the fork axis will move back a little, so increasing the trail. But what affect will this have? What is trail anyway?

Take the ubiquitous shopping trolley. Pull one from any canal or parkland pond and study its front castors. The steering castor angle is pretty much 0 degrees, so the wheel will happily twizzle round any which way with minimum effort and exhibits zero stability. However, the design of the wheel mount introduces a fair bit of trail, which introduces some directional stability to the assembly - push the trolley forward and the wheel will trail the steering axis, so we have some directional control. The fork offset is effectively the distance from the steering bar at the back of the trolley to the castor at the front - pretty huge. So when you want to turn you have to use a fair bit of effort to rotate the trolley. Although the steering is thus quite heavy and ponderous, the relative lack of directional stability of the castor itself means the trolley will happily shoot off in the wrong direction with the minimum of fuss. Heavy steering yet poor stability? ... sounds like an R1! And we all know that a fully laden trolley is harder to manoeuvre than an empty one, just like a heavy bike.

How does this translate to bikes? Well, to improve directional stability they introduce a larger steering angle, typically 24 degrees. Thus the wheel centre is pushed out forward of the steering stem and the pivot point of the tyre on the road is behind the steering axis, so imparting an amount of trail and stabilising the steering, as to turn the wheel the tyre must scrub on the road - hence it is often called the scrub radius. To reduce the trail to an optimum level the forks are offset forward of the steering stem pivot, by typically around 30mm. Ideal steering characteristics are reached by balancing the relationship between the steering angle and the scrub radius. Why not simply provide stable steering by using more trail? Because the increasing scrub radius leads to heavy steering.

Mick demonstrates the art of counter-steering.
Only trouble is, he's using the back wheel.

Initiating a turn on a bike is a little odd, due to the effects of the castor angle of the steering stem, centrifugal torque and gyroscopic forces from the rotating wheel. Whilst travelling straight ahead, if the right bar is pushed forward the wheel will turn to the left, but as it is held out at an angle in front of the bike and the pivot point on the road is thus behind the natural steering axis, centrifugal torque leans the bike to the right. This torque is also trying to lean the front wheel over, which is subject to gyroscopic forces, and these counter the leaning torque by trying to steer the wheel right. All the time that the wheel is turning left, the lean angle of the bike is increasing. The wheel angle slowly reaches zero and then starts to steer right. At this point the lean angle stabilises as the centrifugal torque reverses and balances gravitational forces. Push the right bar now the forces will be reversed and the bike will lift to the upright position. And just as well, or motorcycles would never have caught on.

It's possible to steer a bike without using the bars at all, just by shifting bodyweight and using the hips to initiate a lean angle - to turn right you push your hips left and lean your upper body to the right. However, the forces are weak and so most riders use a combination of steering and body language to steer their bikes. The minimum time taken to transfer the bike from upright to full lean is achieved by counter-steering, shift of bodyweight to the inside of the turn by hanging off, and weighting the footrests. An added benefit from the shift in bodyweight is a reduced lean angle for a given turn, as the centre of mass is moved outward. Useful if you have problems with ground clearance.

An interesting aside. Why is it easy to stay upright on a bike which is moving forwards, but very difficult if it is standing still? It is a commonly proposed theory that the wheels act like gyroscopes and impart the necessary stability to hold the bike upright. Think about this for a minute. If this were true then the faster a bike went the more stable it would become, until it was impossible to lean or steer at all. In fact, a bike is pretty much as stable at 20 mph as at 70 mph. OK, gyroscopic forces on the front wheel do slow steering somewhat at high speed, but the forces involved are insufficient to hold the bike upright. So what's the answer? I'll tell ya - TRAIL. Yup. The friction of the tyres on the road pulls the wheels (don't forget, the rear wheel also has trail) in line behind the steering axis, so the bike maintains line and stability. This stability corrects the minor deviations in balance by allowing the steering to wander about its axis, but always returns the wheels to the inline position. If you've got a steering damper fitted, screw it up to maximum and try and ride your bike ... but be careful, it'll be very unstable as the damper will slow the steering's response to the trail scrub. Or try riding your bike in reverse, when it has negative trail. You won't get far! And don't blame me if you fall off!! If you analyse the steering behaviour when you are riding slowly you will see than the bike doesn't really travel in a straight line, but in a series of shallow curves, first left then right, and this constant gentle weaving allows you to balance the bike. OK, it seems easy to you now, but just think back to when you were learning to ride a bicycle, how long it took you to suss the trick of balancing. Take a look at a trials rider when he is holding his bike on balance at a standstill - he maintains balance by exaggerated shifts in bodyweight and rapid turns of the steering, emulating the natural affect of trail on a moving bike.

OK class, end of physics lesson.