A non-constant holomorphic function $f:X\to Y$ between Riemann Surfaces has a branch point at $p\in X$ if there is no open neighbourhood around p on which $f$ is injective.
I looked up examples and saw that sometimes people only look at the zeroes of the derivative and say that these are the branch points. Why is this the case?
For an analytic function in a neighbourhood of $p$, if $f'(p) \ne 0$ then $f$ is injective in a neighbourhood of $p$, while if $f'(p) = 0$ it is not. Indeed,
$f - f(p)$ has a zero of order $m$ at $p$ if and only if $f'$ has a zero of order $m-1$ there (no zero at all in the case $m=1$). For $w$ in a deleted neighbourhood of $f(p)$, the zeros of $f - w$ in a neighburhood of $p$ are all simple, and the number of them is equal to $m$ by the Argument Principle.