Poisson algebra

A Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation.

Example: Every associative algebra together with the induced Lie bracket $[a,b]=a\cdot b-b\cdot a$. For example, the matrix algebra.
Example: The algebra of functions in a symplectic manifold together with the Poisson bracket induced by the symplectic form. See Poisson manifold and Hamiltonian mechanics.