Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor $\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding.

Given a Scheme $S$, one can consider the category of schemes over $S$. Given an $S$-scheme $X$, one can associate a stack over the category $\text{Sch}/S$, which is an algebraic geometry stack. This gives a functor $\text{Sch}/S\rightarrow \text{A.Stacks}/S$. This is an embedding (I did not check but I am almost sure this is true)

I would like to collect what all notions of differential/algebraic geometry of manifolds/schemes are extended to the setup of differential/algebraic stacks.

I know some notions. For example notion of Cohomology of manifolds is extended to Cohomology of differentiable stacks. Similarly, Cohomology of schemes is extended to Cohomology of algebraic stacks.