and
Another term for dg-manifold .
An N-[supermanifold] is a supermanifold equipped with a lift of the $\mathbb{Z}_2$-grading to a $\mathbb{N}$-grading through the standard homomorphism $even/odd : \mathbb{N} \to \mathbb{Z}_2$.
A Q-[supermanifold] is a supermanifold equipped with an odd-graded vector field $Q$ (i.e. an odd-graded derivation of the algebra of functions) which is homological, i.e. the super Lie bracket with itself vanishes: $[Q,Q] = 0$.
A P-[supermanifold] is a supermanifold equipped with a graded symplectic structure.
It is an old observation by Maxim Kontsevich, amplified by Pavol Severa (ref…) that NQ-supermanifolds are precisely those supermanifolds which are equipped with an action of $End(\mathbb{R}^{0|1})$, the endomorphism monoid of the odd line.
NQ-supermanifolds are an equivalent way of thinking of ∞-Lie algebroids. See the list of references there.
The “Q-manifold”-terminology is due to