The Not So Intuitive Methods of Countersteering & Leaning

In a car one turns the steering wheel to the right to turn right, on a bicycle one pushes the left handlebar away to make a right turn at slow speeds, on a motorcycle under approximately 10 mph the steering is operated like a bicycle, but over approximately 10 mph in order to turn right one needs to press against the right handlebar. This is the phenomenon of countersteering - pushing on the handlebar on the side in which you wish to turn.

The video above explains countersteering visually and is a great resource for new riders. It is also very important to look "through the turn" as many experienced riders would say because it allows your body to position and prepare for what's to come ahead.

Motorcycles are also known for leaning when countersteering. Unlike 4 wheeled vehicles which can change the angle of the front two tires in respect to the back two tires to initiate a turn, in a 2 wheeled vehicle the front tire stays in line with the back tire and the side walls of the tires are used to grip the road.

In a motorcycle lean the rider first repositions his or her seating so that their buttocks is half off of the seat  and the rider positions the head around the area of the mirror in the direction desired to turn. Upon pressing on the handlebar side using countersteering the motorcycle will naturally lean in that directions and the tire sidewalls will take care of the grip to make sure the rider does not crash. The reason for the natural lean is because by positioning the body off-center in the direction desired to turn, the center of gravity is changed more towards the the rider's new body position. In order to stabilize itself, the motorcycle leans initiating the turn.

http://www.michelinmotorcycle.com/index.cfm?event=pilotpower2CT
The Michelin tire company depicts the lean angle capability of one of the model tires they manufacture and is a good example of how far sportbike tire technology has come.