In Hartshorne section $1.1$ he defines affine coordinate ring to be $A/I(Y)$ where $Y\subset \mathbb {A}^n$ is an affine algebraic set and $A=k[x_1,...,x_n]$.
My question is why it involves only affine algebraic sets instead of taking arbitrary subset as $Y$ as $I(Y)$ is defined for each subset of $\mathbb A^n$.
It could be done. However, there isn't a loss of generality because if you define, for an ideal $I\subseteq k[x_1,\cdots,x_n]$, the algebraic set $V(I)=\{x\in \Bbb A^n(k)\,:\, \forall p\in I,\ p(x)=0\}$, then $I(Y)=I(V(I(Y)))$ for all $Y\subseteq \Bbb A^n$. It's similar to the reason why you could define the zero set of a family of polynomials, but you do not lose generality by considering only the case when the family is an ideal.