I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus
- The regular (non-singular) locus can fail to be open.
- Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
- Normalization need not be a module-finite extension.
Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies (and avoid these after some standard operations).
The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as do complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.