moment of inertia
Mass measures how hard it is to get an object moving in a straight line; moment of inertia measures how hard it is to get it spinning about an axis. It is rotation's version of mass. Crucially it depends not just on how much mass there is but on where that mass sits relative to the axis: mass far from the axis resists spinning far more than the same mass near the axis, because it must be swung through a bigger circle. That is why a figure skater spins faster by pulling in her arms, and why a flywheel is built heavy at the rim.
For a single point mass m at distance d from the axis, the moment of inertia is m times d^2 — the mass times the square of its distance. For a continuous body you add up these contributions with a multiple integral: the moment of inertia about an axis is the double or triple integral over the region of (distance from the axis)^2 times the density times the area or volume element. For a planar plate of density rho, the moment of inertia about the x-axis is the integral over R of y^2 rho dA (since y is the distance from the x-axis), about the y-axis it is the integral of x^2 rho dA, and the polar moment about the origin is the integral of (x^2 + y^2) rho dA. Because the distance appears squared, this is also called the second moment of mass. The square is the whole point: it weights faraway mass disproportionately.
Moment of inertia is what enters Newton's law for rotation: torque equals moment of inertia times angular acceleration, the rotational twin of force equals mass times acceleration. It governs how flywheels store energy, how fast a beam or shaft twists under load (where the closely related second moment of area appears in bending and torsion formulas), the swing of a pendulum, and the tumbling of a satellite. Two honest subtleties: moment of inertia is defined relative to a chosen axis, so the same body has different values about different axes; and the parallel-axis theorem lets you shift from the center-of-mass axis to a parallel one by adding (total mass) times (distance between axes)^2, but only between parallel axes.
A uniform disk of mass M and radius a, spun about its central axis, has moment of inertia integral over the disk of r^2 (sigma) dA = (M / 2) a^2, where sigma = M / (pi a^2). The extra r from r dr d-theta combines with the r^2 distance to give an r^3 integrand.
The squared distance plus the polar Jacobian r make the integrand r^3, giving the standard (1/2) M a^2.
Moment of inertia is always about a specified axis — there is no single value for a body. Use the perpendicular distance to that axis (squared), and remember the parallel-axis theorem connects only parallel axes.