the parallel-axis theorem
The moment of inertia of an object depends on which axis it turns about, so in principle you would have to recompute it for every possible axis — a tedious prospect. The parallel-axis theorem is the shortcut that rescues you: once you know the moment of inertia about an axis through the object's centre of mass, you can find it about any parallel axis with a single quick addition.
Precisely, the theorem says I = I_cm + M d^2, where I_cm is the moment of inertia about a parallel axis through the centre of mass, M is the total mass, and d is the distance between the two axes. In words, shifting the axis away from the centre of mass always increases the moment of inertia, by an amount equal to the whole mass treated as if it sat a distance d out. The centre-of-mass axis always gives the smallest possible I.
This is the everyday tool for real rotating objects that do not turn about their centres — a door on its hinge, a rod swung from one end, a pendulum bob on an arm. It is exactly how you set up a physical pendulum, and it saves you from re-deriving a moment of inertia from scratch every time the axis moves.
A uniform rod has I_cm = 1/12 M L^2 about its centre. Swung about one end, the axis shifts by d = L/2, so I = 1/12 M L^2 + M (L/2)^2 = 1/12 M L^2 + 1/4 M L^2 = 1/3 M L^2.
Moving the axis to the rod's end raises its moment of inertia fourfold, from 1/12 to 1/3 M L^2.
The two axes must be parallel, and I_cm must be taken about the axis through the centre of mass; using any other reference axis for I_cm makes the theorem give the wrong answer.