It is easy to show using birational factorization that the only curves on a surface which can be contracted to get an algebraic, smooth surface are smooth $(-1)$-curves. Furthermore, I know of examples of smooth curves of some other negative self-intersection which can be contracted in the algebraic category to result in a singular surface (where the order of the singularity is equal to the negative self-intersection?)
Question 1: Given a curve of negative self-intersection on a complex surface, what is the construction (in the analytic category) of its contraction?
Question 2: Given the curve, what conditions on it that determine whether the contraction is algebraic or not?
Good, clear, elementary references would be fine, as an alternative to an answer!