If we consider the above covariance as a matrix $$C_{ij} =\mathrm{Cov}(X_i,\,X_j),$$ then the diagonal elements are just the familiar variances, $C_{ii} = \mathrm{Cov}(X_i,\,X_i) = \mathrm{Var}(X_i)$. It turns out that all the off-diagonal elements are zero if the stochastic variables are uncorrelated.