Conformal map
A map that preserves both the size and the orientation of angles between curves.
A map is conformal at a point if it preserves the angle between any two curves crossing there, both in magnitude and in orientation (sense of rotation). For a complex function f this happens at every point z_0 where f is analytic and f'(z_0)\neq 0, since near such a point f acts, to first order, like the map "rotate by \arg f'(z_0) and scale by |f'(z_0)|," and rotating-and-scaling is exactly the operation that leaves angles between curves unchanged. Conformality fails wherever f'(z_0)=0, since that local approximation degenerates and angles can be distorted.
Examples
f(z)=e^z is conformal at every point, since f'(z)=e^z is never 0.
f(z)=z^2 fails to be conformal at z=0, where f'(0)=0; it doubles the angle between any two curves crossing there.