Here is a structure theorem of a finite-dimensional noncommutative Poisson algebra A. A nice element epsilon of A will be found, so that the Lie module action of an element of a large Poisson subalgebra of A on A is described in terms of epsilon and the ordinary associative commutator. Consequently, we can figure out a structure of A when the Jacobson radical rad A satisfies (rad A)(2) = 0. This structure theorem leads us to a classification of the finite-dimensional simple Poisson A-modules.