a conformal automorphism of the disk
A conformal automorphism of the disk is a way to reshuffle the unit disk onto itself, smoothly and reversibly, while preserving every angle. Think of it as a rigid motion of the disk — not rigid in the ordinary Euclidean sense (it does distort Euclidean lengths) but rigid in the hyperbolic sense: it is a symmetry of the disk's own non-Euclidean geometry. These are the maps that move the disk around without tearing it, without folding it, and without changing what it 'is' as a geometric object.
Concretely, a map f is a conformal automorphism of the unit disk if it is holomorphic, one-to-one, and onto the disk (its inverse is then automatically holomorphic too). The remarkable classification theorem says every such map has exactly one form: f(z) = e^(i theta) (z - a) / (1 - a-bar z), where a is some point inside the disk (with |a| < 1) and theta is a real angle. Read this in two steps. The factor (z - a)/(1 - a-bar z) is a Blaschke factor: it sends the chosen point a to the center 0 and is itself an automorphism. The factor e^(i theta) is just a rotation about the center. So every disk automorphism is 'pick where to send the center, then rotate' — three real parameters in all (the two coordinates of a, plus the angle theta). The proof that there are no others uses the Schwarz lemma twice: given any automorphism, normalize it to fix 0, conclude it is a rotation by Schwarz, then undo the normalization.
These maps are the symmetries that make the disk a homogeneous model of hyperbolic geometry: they act transitively (you can slide any point to any other) and they are exactly the orientation-preserving isometries of the Poincare metric. They appear everywhere — as the building blocks of the Schwarz-Pick lemma, in the spectral theory of operators on the disk, in hyperbolic tilings, and in complex dynamics. An honest caveat: these are the HOLOMORPHIC (orientation-preserving) automorphisms. If you also allow reflections (anticonformal maps like z -> z-bar followed by an automorphism), you get a larger group, but those reverse orientation and are not holomorphic.
The map f(z) = (z - a)/(1 - a-bar z) with a = 1/2 sends 1/2 to 0 and 0 to -1/2. Check it stays in the disk: when |z| = 1, a short computation gives |f(z)| = 1, so the boundary circle maps to the boundary circle, and by the maximum principle the interior maps to the interior. It is its own kind of mirror-swap of inside-the-disk points.
A Blaschke factor sending a to 0 — the basic non-rotation building block of every disk automorphism.
A frequent confusion: not every Mobius map preserves the disk — only those of the special form above do. A general Mobius map sends circles to circles but may move the unit circle to some other circle entirely. The disk automorphisms are exactly the Mobius maps that happen to keep |z| = 1 fixed as a set and the interior inside.