Let $k$ be a field and $f,g$ be algebraic affine curves in $A_{k}^2$.
1) What does $f,g$ algebraic mean?
I know that a set is algebraic if it is equal to $V(I)$ for some ideal $I$ of the polynomials ring but I don't know what it means for functions.
2) Define the multiplicity of $f$ at $p$ and the interesction multiplicity of $f,g$ at $p$
3) What is meant by the notation $\mathcal{O}_{\mathbb{A}_k^2,O}$
Thanks