A Poisson manifold is a differential manifold M such that the algebra of smooth functions over it, 
is equipped with a bilinear map called the Poisson bracket turning it into a Poisson algebra.
Every symplectic manifold is a Poisson manifold but not vice versa.
A manifold M with a smooth bivector field η can be turned into a Poisson manifold via {f,g}=η(df,dg) provided η(η(df,dg),dh)+η(η(dg,dh),df)+η(η(dh,df),dg) for all f, g, h. For a symplectic manifold, η is nothing other than the inverse of the symplectic form ω, which exists because it is invertible.
See also Poisson supermanifold, Nambu-Poisson manifold