The Hessian is the matrix of mixed partial derivatives:
H=(∂x2∂2f∂x∂y∂2f∂x∂y∂2f∂y2∂2f).
If this matrix has +ve eigenvalues only, then the function is increasing in every direction, and hence we have a minimum. If we have only -ve eigenvalues, then we have a maximum. If the e.vs are mixed, then in some directions the function increases, and in others the function decreases, so we have a saddle point.