An is normal if and only if it is nonsingular.
This statement comes from Kemper, A Course in Commutative Algebra. He says to use Proposition 8.10 and Theorem 14.1.
Theorem 14.1. A Noetherian local ring of dimension one is regular if and only if it is normal.
An irreducible affine curve is normal means that the coordinate ring $K[X]$ is normal. So, by the Proposition, for every $x$, the localization $K[X]_x$ is normal. How can I get that $K[X]_x$ is regular? Is that $K[X]_x$ is a local ring? Could someone tell me ?
An algebraic variety $X$ is regular if all of its local rings $O_{X,x}$ are regular local rings. The latter means that the maximal ideal $M_{X,x}$ of $O_{X,x}$ is generated by $\mathrm{dim}(O_{X,x})$ elements.
Since regularity of a ring is stable under localization it suffices to require that the local rings $O_{X,x}$ in closed points of $X$ are regular.
Regular local rings are normal - in fact they are factorial by a famous theorem due to Auslander and Buchsbaum.
Now for a curve $X$ the dimension of every local ring $O_{X,x}$ in a closed point equals $1$. Hence if $x$ is regular $M_{X,x}$ must have one generator. Theorem 14.1 in combination with the mentioned normality yields the result.