We provide the first social choice theory approach to the question of what constitutes a community in a social network. Inspired by the classic preferences models in social choice theory, we start from an abstract social network framework, called preference networks; these consist of a finite set of members where each member has a total-ranking preference of all members in the set. Within this framework, we develop two complementary approaches to axiomatically study the formation and structures of communities. (1) We apply social choice theory and define communities indirectly by postulating that they are fixed points of a preference aggregation function obeying certain desirable axioms. (2) We directly postulate desirable axioms for communities without reference to preference aggregation, leading to eight natural community axioms. These approaches allow us to formulate and analyze community rules. We prove a taxonomy theorem that provides a structural characterization of the family of community rules that satisfies all eight axioms. The structure is actually quite beautiful: these community rules form a bounded lattice under the natural intersection and union operations. Our structural theorem is complemented with a complexity result: while identifying a community by the most selective rule of the lattice is in P, deciding if a subset is a community by the most comprehensive rule of the lattice is coNP-complete. Our studies also shed light on the limitations of defining community rules solely based on preference aggregation: any aggregation function satisfying Arrow's IIA axiom, or based on commonly used aggregation schemes like the Borda count or generalizations thereof, lead to communities which violate at least one of our community axioms. Finally, we give a polynomial-time rule consistent with seven axioms and weakly satisfying the eighth axiom.