Jochen Heinloth: The Moduli Stack of Vector Bundles on a Curve

We will give an introduction to algebraic stacks, using only basic algebraic geometry. Our main example will be the moduli stack of vector bundles on a projective algebraic curve, which provides a lot of interesting phenomena.
To illustrate the usefulness of the concept, we will compare it to the classical approach using coarse moduli schemes. Here we will see that questions like the computation of cohomological invariants can already be formulated for the stack and tend to become much simpler in this setting. Moreover, these results can be used to deduce information on the coarse moduli space.

To get some impression of what stacks are, there are several introductions to algebraic stacks available, e.g.