Every action meets an equal reaction

Why this rule exists

Forces are never solitary; they are the two ends of a single interaction between two bodies. When A pushes B, B pushes A equally and oppositely, because a force is really an exchange of momentum, and momentum is conserved. The third law is conservation of momentum stated locally. The two forces act on different bodies, which is why they do not simply cancel.

In practice

Identify the pair of interacting bodies and label the two forces — equal, opposite, and of the same type. Never cancel them against each other; they belong on different free-body diagrams. To find recoil, propulsion, or the force underfoot when you walk, follow the reaction onto the other body and apply it there.

Example

A rifle mass M fires a bullet mass m at speed v.

Momentum before = 0   (nothing moving)
Momentum after  = m*v + M*V

Conserved (equal, opposite forces):
0 = m*v + M*V
V = -(m/M)*v   (rifle recoils, slowly)

When it doesn't apply

The instantaneous third law fails when the interaction travels: two moving charges push each other unequally because the field takes time to arrive. Momentum is still conserved once you credit the field itself with carrying the missing momentum.