I am reading a paper, and it uses the following definition:
A morphism ${f : C \to C'}$ of affine curves is the restriction to C of a map of the following form, where ${f_1}$ and ${f_2}$ are elements of ${k[x,y].}$ $${\mathbb{A}^2(k) \to \mathbb{A}^2(k)}$$ $${(x,y)\mapsto (f_1(x,y),f_2(x,y))}$$ If $C$ and ${C'}$ are projective curves, then a map ${f : C \to C'}$ is a morphism if and only if ${C = \bigcup_{i \in I}C_i}$ and ${C' = \bigcup_{j \in J}C'_j}$ for finite sets $I$ and $J$ and open subsets ${C_i,C_j'}$, which are affine curves, such that for all ${i \in I}$, ${f(C_i)\subseteq C_j'}$ for some ${j \in J}$ and ${f|_{C_i}}$ is a morphism of affine curves.
I am very much confused by this definition. Can anyone give an example?
I thought its getting too lengthy in the comments:
Your question boils down to the fact that you consider a morphism of projective curves $C \rightarrow C'$ on open affines in the projective space.
I.e. if $C \subset \mathbf{P}^2$ is a projective curve defined by maybe $YZ^2-X^3=0$ then note that $\mathbf{P}^2$ is covered by three affine charts $U_x,U_y,U_z$ corresponding to the opens where the coordinates $X,Y,Z$ are invertible.
Then $U_x$ is $\mathbf{Spec}(k[Y/X,Z/X])$ and $C \cap U_x$ is defined by $yz^2-1=0$ (affine curve, something you can draw on a sheet of paper), where you set $Y/X=:y$ and $Z/X=:z$. That is one affine patch and the others analogously.