When I asked myself this question a while ago it kept me puzzled and awake at night for quite some time and it doesn't surprise me at all that after so many posts only Paul has touched the essence of the solution to the canopy turning problem by mentioning the pendulum effect.
The BIG difference between an airplane and a canopy is the position of the aerodynamic center in relation to the center of gravity. In an airplane these two are close together. Moving the aileron itself will not turn the plane but only roll it, rolling the lift vector with it creating a sideways pointing component that will turn the plane. The downward deflected aileron on the outside of the turn will create more lift during the roll-in thus, create more induced drag. The upward deflected aileron on the inside of the turn will decrease lift, thus less induced drag. this will yaw the plane to the opposite side (outside) of the turn. This unwanted side effect is called adverse yaw and is compensated for with rudder into the turn during the roll-in and out of the turn during roll-out (Ask your DZ pilot). Note that aileron is used only to roll-in and roll-out of the turn and not during the turn and that the resulting decreased vertical lift component will make the plane decent if not countered with elevator and power.
Under a canopy the center of gravity and the aerodynamic center are some 10 feet apart, depending on the line length. Now, pulling a toggle or rear riser down will increase lift but the weight of the jumper on that extreme long arm will prevent it from rolling. The increased drag however will yaw it, swinging the jumper to the outside (in fact, it's momentum will want to keep it going in a straight line while the canopy turns), rolling the wing and with it the lift vector, creating the sideways force. Note that the "aileron" is used throughout the turn and that the turn is a combination of "adverse" yaw caused by increased drag and rolled lift vector caused by the pendulum effect. Also the decreased vertical component will increase the decent rate. If continued this will develop into a spiral dive which is also confusingly called a stall turn. When pulling down a front-riser the camber of that side will increase, thus increasing drag. The decreased angle of attack counteracts the increased lift due to camber. A gentler turn with less swing and more downward motion is the result. For those of you with CReW canopies, try pulling a B-line. It's much like flying a front-riser turn with pull force comparable to toggles.