In their paper Derived Quot Schemes Kapranov and Ciocane-Fontanine write in the introduction:
Indeed, an algebraic stack is a nonlinear analog of a complex of vector spaces situated in degrees $[-1,0]$ and, for example, the tangent "space" to a stack at a point is a complex of this nature.
Can someone explain this analogy? Pointers to the literature would also be appreciated.
This is a long story and I don't know what the shortest way to tell it is. Here's one version: you can think of an algebraic stack as being, at least locally, presented by a groupoid internal to schemes. E.g. if we consider a quotient stack $X/G$ then the corresponding groupoid is the action groupoid $G \times X \rightrightarrows X$ where the two arrows are the action map resp. the projection; I will stick to this example from now on.
Now what should the "tangent space" to such a thing be? You can imagine taking a point $x \in X$ and considering taking the tangent space at $x$ in the base space $X$ and also at $(e, x)$ in $G \times X$; together we get a groupoid $\mathfrak{g} \times T_x(X) \rightrightarrows T_x(X)$ internal to vector spaces (if we're working in schemes over a field, say), also known as a Baez-Crans $2$-vector space.
Now it is a pleasant exercise to check that the category of groupoids internal to vector spaces is equivalent to the category of two-term chain complexes of vector spaces (and "vector spaces" can be replaced by any abelian category). This is a simplified version of the Dold-Kan correspondence and is explained, for example, in Baez and Crans' Higher-Dimensional Algebra VI: Lie $2$-Algebras, Proposition 8. I think but have not checked carefully that if we apply this correspondence to the above example we should get the two-term chain complex $\mathfrak{g} \to T_x(X)$ where the map is the differential of the map $G \ni g \mapsto gx \in X$ at $g = e$.
More generally we can talk about higher stacks. These look, again at least locally, like simplicial schemes, so their "tangent spaces" look like simplicial vector spaces and these correspond to longer chain complexes by the full Dold-Kan correspondence. The Dold-Kan correspondence generally suggests the intuition that simplicial objects are "nonlinear chain complexes."