The literature on Mandelbrot and Julia sets mentions the phase "critical point" quite a lot, but usually doesn't bother to define what it means.
As best as I can tell, a critical point is just any point where the function's derivative is zero.
This cannot be the whole story, however. For example, the same literature often states that the critical points of $f(z) = z^2 + c$ are "zero and infinity".
Clearly $f'(z) = 2z$. Obviously $f'(0)=0$. So zero is a critical point. But how on Earth do they figure that infinity is a critical point? $f'(\infty) = 2\infty = \infty \not= 0$. (??)
Edit: According to Wikipedia, a critical point is anywhere the function's derivative is zero or undefined.
Is this going to be one of those things involving the complex plane "with a point at infinity"? I'm thinking, perhaps as $z$ approaches $+\infty$ then $2z$ does to $+\infty$, but as $z$ approaches $-\infty$ then $2z$ approaches $-\infty$, which isn't the same limit? Or something like that?