On the complex plane, the set

D = {z: |z| < 1}.
That is, the set of all points whose distance to the origin is strictly less than 1. Note that we do not include the unit circle in the unit disk!

The unit disk is a very common domain in complex analysis and harmonic analysis. It has a compact closure, which makes various questions (such as convergence of a function defined inside the disc to a "nice" function defined on its boundary, the unit circle) interesting.

The Riemann mapping theorem makes the unit circle conformally equivalent to any other simply connected domain, except the entire plane.