If you can ride a motorcycle at above walking speed for more than 100 meters without crashing into anything, then you are already countersteering, but probably just don't realize it. However, you are doing it unconsciously, and there are incalculable benefits in learning how to do it deliberately. These benefits are:
(1) Steering control is more precise, particularly with respect to your track in curves and lane changes.
(2) Much, much quicker response and maneuvering, including accident avoidance.
(3) Much more stable maneuvering.
(4) Much smoother maneuvering.
(5) Much easier two up riding.
For years I wondered why certain people seemed to be able to make their bikes respond so much more quickly than I could, yet with complete stability. The answer is countersteering.
Gyroscopes and such.
At any speed much above walking, IT IS IMPOSSIBLE TO PHYSICALLY TURN THE FRONT WHEEL OF A MOTORCYCLE, the gyroscopic effect of the front wheel will not let you. All turning is accomplished by leaning. Countersteering is all about how to initiate the lean, and maintain it precisely. To countersteer deliberately and effectively, you must understand the principle behind it.
All you need to know are two rules regarding gyroscopes, and how these influence riding a motorcycle. Considering that a motorcycle front wheel is a gyroscope spinning in a vertical plane, the two fundamental rules governing gyroscopes that allow you to steer a motorcycle at speed are as follows:
(1) When a gyroscope spinning in a vertical plane is TILTED, it will TURN to the SAME side to which it is tilted. This is the process by which most people steer, by leaning.
(2) When a gyroscope spinning in a vertical plane is TURNED, it will TILT to the OPPOSITE side to which it is turned. This is the rule that makes countersteering possible.
Now the great majority of people riding motorcycles only deliberately use rule (1) above, i.e. they tilt the wheel (that is they lean the motorcycle) and the wheel then turns to the side it is leaned. Therefore LEAN to the LEFT; turn to the LEFT. They initiate the lean by shifting their body position, or by throwing their weight, although this is also where the unconcious countersteering takes place.
But use of rule (2) allows much better control. Remember the gyroscopic effect will not let you actually turn the wheel, but any attempt to turn it will result in the reaction described in rule (2) above, i.e. if by using the handlebars you attempt to turn the wheel to the RIGHT, the wheel and hence the bike will tilt in the OPPOSITE direction, to the LEFT. As soon as it tilts, rule number one comes into play and the bike will turn to the side it is leaning. In other words, try to turn the handle bars to the RIGHT; the bike will lean LEFT and turn LEFT. The bar won't actually turn, the gyroscopic effect wont let it, just the attempt to turn it will accomplish this effect. The harder you try to turn, the more pronounced the effect will be.
So to turn LEFT on a moving motorcycle, you push the LEFT handlebar gently but positively forward. It will not perceptibly move, but the bike will instantly lean LEFT and turn to the LEFT. A hard push forward on the left bar, and the bike will literally dive into a leaning left turn. You keep the pressure on through the turn, modulating the amount of pressure regulates the lean and hence the radius of the turn. Relieve pressure and the bike straightens up immediately, and tracks straight. Slight pressure forward on the right bar then turns you to the right, as in a lane change or accident avoidance maneuver.
Do not move your body through this maneuver, just sit the motorcycle. The handle bar pressure creates the lean, your body will just lean in perfect line with the bike. So will your passenger's. Enjoy the ride.