center of mass
The center of mass is the single average point where you can imagine all of an object's mass to be concentrated — its balance point. Hold a hammer and find the one spot where it balances on your finger; that is essentially it. For a whole system of objects, the center of mass is the mass-weighted average of all their positions: heavier pieces pull the point toward themselves.
For particles, x_cm = (m1 x1 + m2 x2 + ...) / (m1 + m2 + ...), and likewise for the y and z coordinates; a big mass sways the average far more than a small one. The reason this point is so useful is a beautiful theorem: the center of mass of any system moves exactly as if all the mass were a single particle sitting there, pushed only by the net EXTERNAL force. Internal forces — however violent — never move it.
That is why a wrench tossed spinning across a table drifts in a smooth straight line at its center of mass while its ends whirl around, and why a firework's fragments fly apart yet their center of mass keeps sailing along the original arc. The center of mass is the 'honest' position of a complicated system — the thing Newton's laws really talk about.
Two skaters, 50 kg and 100 kg, hold a rope 3.0 m apart. Put the 50 kg one at position 0 and the 100 kg one at 3.0 m: x_cm = (50 times 0 + 100 times 3.0) / 150 = 2.0 m. The balance point sits twice as close to the heavier skater. When they reel the rope in, they meet exactly there — the center of mass does not move, because the pull is internal.
The center of mass leans toward the heavier body and stays put under purely internal forces.
The center of mass need not lie inside the object at all — for a ring or a boomerang it sits in the empty space in the middle. It is also distinct from the center of gravity, though the two coincide whenever gravity is uniform across the object, as it is in everyday life.