In
complex analysis a domain is a
connected open subset of the
complex numbers
C.
In complex analysis we mainly study functions defined on connected open subsets of C, which is the reason for giving such sets a special name.
We want a domain to be open because while a function may be differentantiable at a single point or at all points in some arbitrary (non-open) set, requiring that a function is differentiable in an open set is a much stronger condition. Such a function is called analytic.
The condition that the set be connected is mainly to avoid trivialities. If a domain could be a union of disjoint sets then a function defined in a domain have quite different properties in the different components, i.e. we could define a function to map z to z2 on one component, and constant on another component. Since we want to be able to make statements about the properties of analytic functions throughout the set where they are defined we therefore tend to restrict our attention to functions defined on connected sets.