self-adjoint operator
A self-adjoint operator is one that equals its own adjoint: T = T^*. In coordinates with an orthonormal basis this means the matrix equals its conjugate transpose (Hermitian), or in the real case its plain transpose (symmetric). These are the best-behaved operators in all of linear algebra.
The defining identity <T u, v> = <u, T v> for all u,v makes the quadratic form <T v, v> real for every v (since it equals its own conjugate). From this two gifts follow: every eigenvalue is real, and eigenvectors for distinct eigenvalues are automatically orthogonal. Both are short, satisfying proofs you should do once by hand.
The crown is the spectral theorem: a self-adjoint operator on a finite-dimensional space has an orthonormal basis of eigenvectors and is therefore orthogonally (unitarily) diagonalizable with real eigenvalues. Symmetric real matrices and Hermitian complex matrices are exactly the diagonalizable-by-rotation matrices.
Self-adjointness is the abstract face of symmetry, and it is everywhere: the Hessian of a smooth function, the covariance matrix of a random vector, the Hamiltonian of a quantum system. Wherever a real spectrum and orthogonal eigenstructure are demanded, self-adjointness is the hypothesis doing the work.
The hallmark consequences of self-adjointness: a real spectrum and an orthogonal eigenbasis.
Self-adjoint is stronger than normal but a special case of it: T = T^* certainly commutes with T^*. The spectral theorem applies to all normal operators; for self-adjoint ones it additionally guarantees the eigenvalues are real.