Counter steering is the name given to the "counter-intuitive" technique used by motorcyclists to turn corners.
Motorcycles turn corners using the gyroscopic roll reaction force of the spinning front wheel. The attitude of the bike rolls about a longitudinal axis thru the center of mass. Once the roll attitude angle has been achieved, the bike will remain there until a new steering command is input.
Most folks may not care about the physics, and certainly don't need to know any math in order to ride a motorcycle. So how about a description of counter steering that appeals to common sense and intuition?
When riding a bicycle or a motorcycle, counter steering is a method of preparing for a turn by a small, momentary turn of the handlebars in the opposite direction. The necessity to counter steer becomes noticeable above a minimum speed. A bike can negotiate a curve only when it leans towards the inside of the turn, at a camber angle appropriate for the velocity and the sharpness of the turn. At higher velocities, the bike has a strong natural tendency to remain upright.
The transition of riding in a straight line to negotiating a turn is a process of shifting the balance of the bike. In a sense, the bike needs to be thrown off balance first.
If a biker wants to turn to the right, he first throws the bike off balance by a well-timed jolt to the handlebars, momentarily pointing the front wheel slightly to the left.
As the bike is being tipped over to the right (by centrifugal forces) the front wheel is steered into the curve, and the curve is negotiated with the proper inward lean. This process requires little physical effort, because the geometry of the steering of a bike is designed in such a way that the front wheel has a strong tendency to follow the balance of the bike.
Counter steering is indispensable for motorcycle steering. Most people are not consciously aware that they employ counter steering when riding their bicycle. Their body has learned to include the well timed counter steering jolt, they have learned to do so while learning to ride a bicycle in childhood. Often people simply assume that the steering of a bicycle is just like the steering of a car. Their subconscious balance skills know better: they shift their balance first, and then the front wheel tends to steer into the turn that the rider has in mind.
Even more so than on a bicycle, mastering the technique of counter steering is essential for safe motorcycle riding. Above around 10 mph, it is not possible to conventionally steer a motorcycle and counter steering is used exclusively to change direction. Much of the art of motorcycle cornering is learning how to effectively "push" the grips into corners and how to maintain proper lean angles through the turn.