For $\mathbb{P}^n$ we can let $U_i = \{(x_1:\cdots:x_i:\cdots:x_{n+1}) : x_i \neq 0\}$. Then let $C \subset \mathbb{P}^n$ be a projective plane curve. We can decompose $C$ into a union of affine plane curves:
\begin{equation} C = \bigcup\limits_{i=1}^{n+1} C_i \end{equation}
where $C_i = C \cap U_i$.
But it's stated in Milne's Elliptic Curves, page 13, that for the curve $C: Y^2 Z = X^3 + aXZ^2 + bZ^3$, the covering is $C = C_1 \cup C_2$. Why is $C_0$ not there? Is it empty? Or do we have $C_0 \subset C_1 \cup C_2$?
Each point not lying in $U_1\cup U_2$ must have $x_1=0$ and $x_2=0$, so it is equal to $[x_0:x_1:x_2]=[1:0:0]$. In your equation, the point $[X:Y:Z]=[1:0:0]$ does not belong to the curve (because of the $X^3$), so you only need $C_1$ and $C_2$.