The balance point of a system
Balance a ruler on one finger and you find its center of mass — the point where the mass is, in a sense, evenly distributed around. For a collection of particles, it is the average of their positions, but weighted by mass so heavier particles pull it toward themselves.
The center of mass: the mass-weighted average position of all the particles, where M is the total mass.
Quick example: put a 1\text{ kg} mass at x = 0 and a 3\text{ kg} mass at x = 4\text{ m}. The center of mass sits at x_{\text{cm}} = (1\cdot 0 + 3\cdot 4)/(1+3) = 12/4 = 3\text{ m} — three-quarters of the way toward the heavier mass, exactly as balance intuition predicts.
How the center of mass moves
Here is why the center of mass is more than a geometric curiosity. Its velocity is just the total momentum divided by the total mass — so the total momentum of any system is exactly as if all its mass were concentrated at the center of mass, moving as one particle.
The center of mass obeys Newton's second law for the whole system: it accelerates only under the net external force, blind to all internal forces.
One more tool for experts: view a collision from the center-of-mass frame, the reference frame moving with \vec{v}_{\text{cm}}. In that frame the total momentum is zero, so the two objects always approach and (if elastic) leave with equal-and-opposite momenta. Many collision problems become almost trivial there and are then transformed back to the lab frame.
Two tools together: the ballistic pendulum
The finest problems combine momentum with energy, using each where it is valid. The classic ballistic pendulum measures a bullet's speed by firing it into a hanging block and watching how high the block swings. The trick is knowing which conservation law applies to which stage — a mistake here is the single most common error students make.
The ballistic pendulum: momentum conservation for the impact, then energy conservation for the swing. Using the wrong law in either stage gives a wrong answer.
Where this leads next
Momentum is one of the deepest ideas in physics, and you now hold the linear version. Next, rotation has its own conserved partner — angular momentum — that explains spinning skaters and stable planets. In energy and momentum together you have the two master conservation laws of mechanics. And at speeds near light, momentum must be redefined; Einstein's relativistic momentum grows without bound as an object approaches c, which is exactly why nothing massive can ever reach it.