I would like to understand in an intuitive level first and then a technical level also (keeping in mind I am a physicist) the difference between the moduli space of vector bundles and the moduli stack of vector bundles over an algebraic variety or scheme over a complete field, in specific $\mathbb{C}$ is sufficient for me.
We can assume that the rank of the vector bundle is fixed. There is some jargon that I cannot figure out: moduli space vs. coarse moduli space, moduli space vs. moduli stack, moduli functor and moduli space etc.
I also want to note that I understand to some extent the definition of an algebraic stack as a 2-category but I do not see how this can be a possibly singular variety whose points parametrize isomorphism classes of vector bundles.
I also am a bit confused on the same question with the replacement of vector bundles to sheaves. But I do have some idea about the differences between the moduli space of the first and the moduli space of coherent torsion free sheaves.
To summarize:
- Can you please explain in both intuitive and (Semi)-technical level the difference between the words space and stack?
- Can you clarify when we need one and when the other?
- Can you explain the jargon?
- What happens if we switch to vector bundles to sheaves?
P.S. There are some nice books like the one of Huybrechts but it is quite above my level for now.