Geometric, algebraic, and homological properties of Poisson structures on smooth manifolds are studied. Noncommutative (NC) foundations of these structures are introduced for associative Poisson algebras; noncommutative generalizations of such notions of the classical symplectic geometry as a degenerate Poisson structure, a Poisson submanifold, a symplectic foliation, and a symplectic leaf for associative Poisson algebras are also introduced. These structures are considered for the case of the endomorphism algebra of a vector bundle, and a full description of the family of Poisson structures for this algebra is given. An algebraic construction of the reduction procedure for degenerate noncommutative Poisson structures is developed. An NC generalization of the Bott connection on a foliated manifold is introduced by using the notions of NC submanifold and quotient manifold. This definition is applied to degenerate NC Poisson algebras, which allows one to consider the Bott connection not only for regular but also for singular Poisson structures.