The main purpose of the present article is to introduce the notion of Poisson structures on an associative algebra, and to discuss several examples. In the first section, the author recalls the \(G\)-brackets for associative algebras which generalize the usual Schouten-brackets for multivector fields in differential geometry. These brackets are essential to the introduction of noncommutative Poisson algebras. Some formulas related to Lie derivatives and contradictions in the noncommutative setting are also obtained. In the second section, the author introduces the notion of Poisson structures on an associative algebra, and then studies basic properties. He proves that the center of a noncommutative Poisson algebra is a Poisson algebra in the usual sense. Poisson cohomology and one- differential Chevalley cohomology are introduced for noncommutative Poisson algebras. One of the main purposes of introducing Poisson structures on associative algebras is to study Poisson reduction for some badly behaved Poisson actions. Section 3 is devoted to the discussion in this direction. He replaces the usual quotient spaces by some noncommutative algebras. This idea has been already used in noncommutative differential geometry mentioned by the author in the references. In Section 4, the author introduces and studies a canonical Poisson structure on noncommutative two-torus and computes its corresponding Poisson cohomology.