the inertia tensor
The inertia tensor is the rotational counterpart of mass, but with a twist: it is a 3x3 matrix rather than a single number, because how hard a body is to spin up depends on the axis. A long rod is easy to spin about its length but hard to spin end-over-end; one object, but different resistances in different directions. The inertia tensor packages all of that directional information into one object.
For a body of density rho(r), its components about a chosen origin are I_ij = integral of rho (r^2 delta_ij - x_i x_j) dV, or for point masses I_ij = sum over alpha of m_alpha (r_alpha^2 delta_ij - x_i,alpha x_j,alpha). The diagonal entries I_xx, I_yy, I_zz are the familiar moments of inertia about the axes; the off-diagonal entries are the products of inertia. Its whole reason for existing is the relation L = I omega -- angular momentum equals the inertia tensor acting on angular velocity. Because I is a matrix, L is in general NOT parallel to omega. The rotational kinetic energy is T = (1/2) omega . I . omega = (1/2) I_ij omega_i omega_j.
The inertia tensor is real and symmetric (I_ij = I_ji), and positive definite, which guarantees it can be diagonalized -- that is the origin of the principal axes. It depends on the choice of origin: shift the origin and it changes according to the parallel-axis theorem, I_ij(new) = I_ij(cm) + M(d^2 delta_ij - d_i d_j). You meet it wherever a body's angular momentum and angular velocity point in different directions: wobbling shafts, tumbling asteroids, unbalanced wheels.
For a uniform solid sphere of mass M and radius R about its center, the inertia tensor is isotropic: every diagonal entry equals (2/5) M R^2 and all products of inertia vanish, so L = I omega is parallel to omega about any axis. A sphere spins the same way no matter how you orient it.
A sphere has a trivial (multiple-of-identity) inertia tensor; most bodies do not.
A common slip is to write L = I omega with a scalar I, which is only valid when omega happens to lie along a principal axis. In general angular momentum and angular velocity are not parallel, and that is precisely why the inertia tensor is a tensor and not a number.