Static Friction

A heavy box sitting still has forces on it that must sum to zero, so if you push it with 15 N and it doesn't move, something is pushing back with exactly 15 N. That something is static friction. It isn't a fixed force like gravity. It's a response force: it grows to match whatever you apply, so the box stays put. This only works up to a point. Once your push exceeds the maximum the surfaces can supply, friction can't keep up anymore, and the box breaks free and starts sliding.

The maximum value static friction can reach is f_s,max = mu_s times N, where mu_s is the coefficient of static friction and N is the normal force. This formula gives you the ceiling, not the actual friction force at any given moment. Below that ceiling, static friction simply equals whatever force is trying to cause sliding, in the opposite direction, so the object stays in equilibrium. That's why a box being pushed gently has a small static friction force even though mu_s times N might be much larger.

This self-adjusting behavior connects static friction to everything else in force analysis: you find it the same way you find any unknown force in an equilibrium problem, by applying Newton's second law with acceleration equal to zero. It also explains something that looks backwards at first: static friction between tires and road is what pushes a car forward. The spinning tire pushes backward on the road, and by Newton's third law the road pushes forward on the tire, a force made possible only because static friction prevents the tire from slipping.