A Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (let me clarify a bit. M isn't a point set space and so, doesn't "really" exist, and really, this algebra is all we have), 
is equipped with a bilinear map called the Poisson superbracket turning it into a Poisson superalgebra.
Every symplectic supermanifold is a Poisson supermanifold but not vice versa.
See also Poisson manifold, Poisson algebra, Poisson superalgebra, noncommutative geometry