Summary: In this paper, we completely characterize the normal Toeplitz operator and commutator of two Toeplitz operators with conjugate holomorphic or holomorphic symbols on the Dirichlet space of the polydisk. We show that the two Toeplitz operators with conjugate holomorphic symbols are commutative on the Dirichlet space of polydisk if and only if two symbols are linearly dependent, and also prove that Toeplitz operator with holomorphic symbol and Toeplitz operator with conjugate holomorphic symbol are commutative if and only if one of the two symbols is a constant.