Let $k$ be an algebraically closed field, and let $B$ be a finitely generated $k$ algebra that is also a Domain. Then $B$ is the affine coordinate ring of some affine variety $Y$; this part is straight out of Hartshorne and is not terribly difficult to understand. However, I am having trouble making sense of the following extension of this statement.
If $B$ is Dedekind, then $Y$ is both non-singular and has dimension 1.
This shows up in Hartshorne (page 41, Lemma 6.5, last paragraph).