What is meant by the statement that an operator is Hermitian?

Consider a particle of mass $m$ in a real potential $V(x)$ in one dimension. Show that the Hamiltonian of the system is Hermitian.

Starting from the time-dependent Schrödinger equation, show that

$$\frac{d}{d t}\langle\hat{x}\rangle=\frac{1}{m}\langle\hat{p}\rangle, \quad \frac{d}{d t}\langle\hat{p}\rangle=-\left\langle V^{\prime}(\hat{x})\right\rangle$$

where $\hat{p}$ is the momentum operator and $\langle\hat{A}\rangle$ denotes the expectation value of the operator $\hat{A}$.