the lever arm
The lever arm is the reason a long wrench beats a short one and a door opens more easily at its edge. It is the effective distance at which a force does its turning — the leverage. Give a force a bigger lever arm and the same push produces a bigger twist; give it none and even a huge force produces no turning at all.
Precisely, the lever arm is the perpendicular distance from the axis of rotation to the line along which the force acts. Torque is then simply this lever arm times the force: tau = (lever arm) x F, which is the same as tau = r F sin theta, where r sin theta is exactly that perpendicular distance. The clever part is the perpendicular: only the part of the geometry at right angles to the force counts. Push a wrench straight in line with the bolt and the lever arm is zero, so the torque is zero.
This one idea explains a whole family of tools and tricks: crowbars, bottle openers, door handles placed far from hinges, and why you instinctively grab the far end of a wrench. To get the most turning, apply the force as far from the axis as you can and at right angles to the line joining them.
You grip a 1.2 m wrench and push, but your force makes a 30 degree angle with the handle. The lever arm is r sin theta = 1.2 x sin30 = 0.6 m — only half the handle's length is doing useful turning.
The lever arm is the perpendicular distance, not just the distance to where you push.
The lever arm is the perpendicular distance to the force's line of action, not simply the distance to the point of contact; a force pointed at the axis has zero lever arm.