On Toni Annala - Resolution of Singularities, I read that "as $y^2 - x^3 - x^2$ is an irreducible polynomial, the coordinate ring $O_C (C)$ is an integral domain, as is the localization $O_{C, p}$ at origin p". Where C is the nodal curve defined as the solution of $y^2 - x^3 - x^2$.
What is the relationship between irreducibility of a polynomial and integral domain?
I did not understand the concept of localization in this context.
If $R$ is a UFD, then so is $R[x]$. Therefore also $k[x,y]$ is a UFD for any field $k$. If we are now given an irreducible polynomial $f \in k[x,y]$, then it is also a prime element (since we are in a UFD) and thus generates a prime ideal. Therefore the quotient ring $k[x,y]/(f)$ is an integral domain. This domain is also the coordinate ring of the affine variety $V(f)$ defined by $f$. This answers the first question.
For the local ring at the origin, note that any localization of a domain $R$ is still contained in the field of fractions of $R$. As subrings of this domain, every localization of a domain once again is domain. Now we just need to link this to the given variety $X = V(y^2-x^3-x^2)$. Here the local ring $\mathcal{O}_{X,p}$ at some point $p \in X$ is the localization of of the coordinate ring $k[x,y]/(y^2-x^3-x^2)$ at the prime ideal which defines the point $p$. Therefore we are in the previously described situation.
Since you were mentioning that you do not understand localizations in this context, I think you should maybe have a look at some commutative algebra book to complement your notes on resolutions of singularities. Maybe also a book on basic algebraic geometry as Kreiser also just proposed, since these are elementary things.