A central notion in many algebro-geometrical stuff appears to be so-called "representable morphisms". A general (read: hand-wavy) definition could be the following, as far as I can tell:
Let $\mathsf{C}$ be a site, and let $f : F \to G$ be a morphisms of sheaves over $\mathsf{C}$. Then $f$ is representable by an object of type $T$ if, for all representable sheaves $h_X$ and all morphisms $h_X \to G$, the fiber product $F \times_G h_X$ is of type $T$. (Here the type $T$ could be "affine scheme", "scheme", "algebraic space", "stack", "manifold", "differentiable manifold", "open subset of $\mathbb{R}^n$"...)
I'm more used to topology, so of course my first instinct was to try and see what this meant for topological spaces. As far as I can tell, a continuous map $f : X \to Y$ is representable by an open subset of $\mathbb{R}^n$ (resp. a manifold) iff for all open subsets of $Y$ that are homeomorphic to an open subset of $\mathbb{R}^n$, then $f^{-1}(U)$ is an open subset of $X$ homeomorphic to an open subset of $\mathbb{R}^n$ (resp. to a manifold).
Honestly, this isn't very enlightening... I can follow the definition and understand how the "representability" condition is used, but I have no real intuition for it. What does it mean? Surely the name wasn't chosen randomly, so if a morphism is represented by (say) a scheme, then what scheme is that, and in what sense does it represent $f$?