The set of all points at unit distance from the origin. In the complex plane, this is {z: |z| = 1}. Astute geometricians will realise that this is indeed the locus of points equidistant from a focus, a set vulgarly referred to as a "circle".
The unit circle is the boundary of the unit disk (of which, however, it is not a part). It is often used to formulate theorems of complex analysis, due to its utmost symmetry (it is, after all, a circle) and its uniticity.