We study the decomposability and the subdifferential of the tensor nuclear norm. Both concepts are well understood and widely applied in matrices but remain unclear for higher-order tensors. We show that the tensor nuclear norm admits a full decomposability over specific subspaces and determine the largest possible subspaces that allow the full decomposability. We derive novel inclusions of the subdifferential of the tensor nuclear norm and study its subgradients in a variety of subspaces of interest. All the results hold for tensors of an arbitrary order. As an immediate application, we establish the statistical performance of the tensor robust principal component analysis, the first such result for tensors of an arbitrary order.