What is meant by the statement than an operator is hermitian?

A particle of mass $m$ moves in the real potential $V(x)$ in one dimension. Show that the Hamiltonian of the system is hermitian.

Show that

$$\begin{aligned} \frac{d}{d t}\langle x\rangle &=\frac{1}{m}\langle p\rangle \\ \frac{d}{d t}\langle p\rangle &=\left\langle-V^{\prime}(x)\right\rangle \end{aligned}$$

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