I saw this general construction of a line bundle in the book Integrable Systems: Twistors, Loop Groups and Riemann Surfaces by Hitchin, Segal and Ward (Chapter 2 Section 1 before Definition 1.5).
Take a point $p\in M$ and a chart $(U_0,\phi)$ such that $\phi(p) = 0$. Denote the local coordinates given by this chart by $z$. Let $U_1 = M\setminus\{p\}$. Then the transition function $g_{01}(q) = \phi(q)$ for $q\in U_0\cap U_1$ gives a line bundle $L_p$ on $M$ by gluing $U_0\times\mathbb{C}$ and $U_1\times\mathbb{C}$ over $U_0\cap U_1$ with the identification $$ U_0\times\mathbb{C}\ni (q,w)\sim(q,g_{01}(q)w)\in U_1\times\mathbb{C}\, . $$
For example, if we replace $\mathbb{C}$ with $\mathbb{R}$, and let $M$ be the $1$-sphere $S^1$, then the above construction gives the Mobius bundle (I think).
Now let $M= \mathbb{C}P^1 = S^2$, $p=0$ and $U_0$ be the unit disc containing $p$. Using the standard chart map on $\mathbb{C}P^1$, what does the line bundle $L_p$ look like?