This question is well enough described in a any QM textbook. You do not make much effort to look there. In simple, Hermitian operator is the observable represetation, or if not rigorously speaking, it reflects the measurement procedure in a some quantum state. For example, let we have an spin-up directed electron state [itex]\left | + \right>[/itex]. Measurement of the z-directed spin by [itex]\hat S_z[/itex] in this state is reflected in the equality [itex]\hat S_z \left | + \right> = +\hbar/2 \left | + \right> [/itex]. This mean the result you will get is [itex]+\hbar/2[/itex]. A key feature of a Hermitian operator is real numbers of their eigenvalues.