Newton's Third Law
Newton's third law states that for every action force there is an equal and opposite reaction force acting on the other object.
Definition
Newton's third law says that forces always come in pairs: when you push on something, it pushes back on you just as hard, in the opposite direction. Stated carefully, when object A exerts a force on object B, object B exerts a force of equal magnitude and opposite direction on object A; the two forces act on different objects, so they never cancel each other, and every force in the universe is one half of such a pair. In symbols, $\vec{F}_{AB} = -\vec{F}_{BA}$, where $\vec{F}_{AB}$ is the force on A due to B. Because interaction forces are equal and opposite, the internal forces within any system sum to zero, which is why the total momentum of an isolated system is conserved. The third law holds exactly for contact forces and gravitation; for electromagnetic forces between moving charges, the field itself must be included for momentum to balance.
Example
When you jump, your legs push down on the ground and the ground pushes up on you, and that upward push is what sends you into the air. A swimmer who pushes backward on the water with $150\ \text{N}$ is pushed forward by the water with $150\ \text{N}$: the swimmer accelerates forward and the water is pushed backward. On ice, a $60\ \text{kg}$ skater pushes a $90\ \text{kg}$ skater with $180\ \text{N}$ for $0.5\ \text{s}$, so each receives an impulse of $90\ \text{N}\cdot\text{s}$. The lighter skater moves off at $90 \div 60 = 1.5\ \text{m/s}$ and the heavier one at $90 \div 90 = 1.0\ \text{m/s}$ in the opposite direction.
Key Insight
You cannot touch without being touched: punch a wall and the wall punches your hand back with exactly the same force, which is why it hurts. A rocket does not push against the air; it pushes hot gas backward, and the gas pushes the rocket forward, which is why rockets work in the vacuum of space. Equal forces do not mean equal effects, though. A bug and a windshield exert equal forces on each other, but the bug's tiny mass gives it an enormous acceleration while the car barely notices.