Without getting into definitions, we have an intuitive sense of what a "community" is. The Louvain Method for Community Detection is one of the best known mathematical techniques designed to detect communities.
This method requires typical graph data in which people are nodes and edges are their connections. It's easy to imagine this data in the context of Facebook or LinkedIn but the technique applies just as well to any other dataset like cellular phone calling records or pen-pals.
The Louvain Method provides a means of measuring the strength of any proposed community based on a concept known as Modularity. Modularity is a value in the range that measure the density of links internal to a community against links external to the community. The quite palatable assumption here is that a genuine community would have members that are strongly interconnected.
A community is not necessarily the same thing as a clique; it is not required that all community members know each other. Rather, we simply define a community as a graph structure where the nodes are more connected to each other than connected to people outside the community.
It's only natural that any person in a community has many connections to people outside that community. The more a community has internal connections over external connections, the stronger that community is considered to be. The Louvain Method elegantly captures this intuitively desirable quality.