The density matrix (density operator) in quantum mechanics is defined as $$\hat{\rho} = \sum_{i} p_i |\psi_i\rangle \langle \psi_i|\, ,$$ where the $|\psi_i\rangle$ are a full orthonormal system and the $p_i$ are probabilities, so $$p_i \geq 0, \quad \sum_i p_i =1 \, . $$
When is the density matrix not diagonal? If we express the matrix in the basis $|\psi_i\rangle$, shouldn't it be always diagonal?