In Quantum Mechanics, the hermitian adjoint of an operator A is denoted A+, and is defined to be the operator that satisfies
    <x|A|y> = <y|A+|x>*
for all states labeled by x and y. By taking these labels to be discrete, the matrix Aij = <i|A|j> can be said to be hermitian if its transpose is equal to its complex conjugate.

Hermitian operators are said to correspond to observables (that is, their eigenvalues are the quantities that are measured by experiment), which is awfully handy since their eigenvalues are all real.