An operator need not be hermitian. For instance, the harmonic oscillator creation operator $\hat a^\dagger$ is not hermitian, and neither is the angular momentum lowering operator $\hat L_-$. Yet both are perfectly legitimate (linear) operators, i.e. they act linearly on a state and produce a different state.
Setting aside subtle points about domains of operators and self-adjointness, observables must be hermitian (in the sense that their matrix representations are hermitian matrices) because eigenvalues of hermitian matrices are real, which is good since in a lab we measure real (rather than complex) quantities. Moreover, hermitian matrices have a complete set of eigenvectors that spans the entire space.
Note that it is important to realize that this doesn’t imply that non-hermitian operators cannot have eigenvalues or eigenvectors, just that there’s no guarantee the eigenvalues are real and the eigenvectors for a complete set. The usual example of this is the harmonic oscillator coherent state $\vert \alpha\rangle$ (where $\alpha$ is any complex number) which is an eigenvector of the annihilation operator $\hat a$, with complex eigenvalue $\alpha$. The eigenvalue need NOT be real since $\alpha$ can be complex, and the coherent states form an overcomplete set of vectors for the Hilbert space of the harmonic oscillator.