TITLE: (geometric/intuitive) interpretation of ext
QUESTION [11 upvotes]: In my current work I have to deal a lot with ext-groups (of modules). I feel kind of familar with the formalism, e.g. the connection between n-th extensions and ext. 
But I don't have a feeling about the meaning of $Ext$. Is there a informal/intuitive (geometric) interpretation of $Ext(M,N) $ in terms of Morphisms $M \to N$?
For example, how far is the following interpretation away from beeing right?  

Consider a free resolution $\cdots \to F_n \to \cdots \to F_1 \to F_0 \to M$ of $M$ does $Ext^i(M,N)$ tells me something about the morphisms in the i-th syzygy $M_i$? e.g consists $Ext^1(M,N)$ of the morphisms of the module generated by the relations of the generators of $M$ modulo the ones, which come from the trivial relations?

I'm mostly interested in the case of $\mathcal{O}_X$-Modules for (toric) varieties or $\mathbb{C}[S]$-Algebras for a semi-groups $S$.
best regards,
Johannes

REPLY [4 votes]: "Is there a informal/intuitive (geometric) interpretation of $\text{Ext}(M,N)$ in terms of Morphisms $M \longrightarrow N$?"
There is an interpretation of $\operatorname{Ext}$ groups exactly as some kind of morphisms $M \to N$. For this you consider instead of abelian category $A$ (of modules or sheaves of modules) derived category $D(A)$, then 
$$
\text{Ext}^i(M,N)=\text{Hom}_{D(A)}(M,N[i]).
$$
This is nothing more than interpretation of $\operatorname{Ext}$ in terms of resolutions, but sometimes it could simplifies calculations (for example, definition of product is obvious)  and it looks (at least for me) like more conceptually clear definition.