I am relatively new to ring theory, though I have an idea of what things kind of are. I have been learning about Dedekind domains, which are integral domains which are Noetherian, integrally closed and where every prime ideal is maximal. I have also heard that for non-singular curves such as $y^2 = x^3 - x$, the coordinate ring $k[x, y] / \langle y^2 - x^3 + x \rangle$ is a Dedekind domain, where the conditions of not having zero divisors roughly correponds to not having nodes, integrally closed corresponds to not having nodes, and Noetherian corresponds to things being finite (I believe this is related to divisors? Which I have heard of but do not understand well enough).
I tried looking up references on this, but it seems that people mostly state this as a "see this interesting correspondence!" example to ring theory beginners. Therefore, I was wondering if anyone could make this more specific, i.e. how the algebraic geometry concepts coincide with the purely algebraic side, especially the condition of being integrally closed, as it seems completely random to me from the algebraic perspective as well. Any specific references will also be great (for ring theory but also for the geometry).
Also a slightly different question, but can someone provide an example of an integral domain which is Noetherian and with Krull dimension $1$, but does not have unique prime ideal factorisation? As I mentioned, I can't "conceptually understand" this condition at all.
Thanks in advance!