normal operator
A normal operator is one that commutes with its own adjoint: T T^* = T^* T. The condition looks technical, but it pinpoints exactly the class of operators for which the spectral theorem holds in full: T is normal if and only if it has an orthonormal basis of eigenvectors (over C).
Self-adjoint operators (T = T^*) are normal, unitary operators (T^* = T^-1) are normal, skew-adjoint ones (T^* = -T) are normal — they are all special cases, distinguished only by where their eigenvalues sit in the complex plane (real axis, unit circle, imaginary axis). Normality is the umbrella that unifies them.
Why does commuting with T^* matter? Because it forces ||T v|| = ||T^* v|| for every v, which makes eigenvectors of T also eigenvectors of T^* (with conjugate eigenvalue). That symmetric grip on both T and T^* is precisely what allows simultaneous orthonormal diagonalization.
The boundary is sharp: a non-normal matrix, say a single nontrivial Jordan block, may not be diagonalizable at all, and even when it is, its eigenvectors need not be orthogonal. Normality is the exact dividing line between operators a unitary change of basis can fully tame and those it cannot.
Normality is exactly unitary diagonalizability: commuting with the adjoint is equivalent to an orthonormal eigenbasis.
Litmus: T is normal exactly when ||T v|| = ||T^* v|| for all v. The complex spectral theorem is then T = U D U^* with U unitary and D diagonal (possibly complex entries).