As You said, there a lot of ways to define communities; one possible definition of is: Sets of nodes with lots of connections inside and few to outside (the rest of the network). So a natural intuitive measure is Conductance (normalized cut) that is (# edges cut / # edges inside). so the best community is that one with a lower Conductance. You can find an interesting paper at the attached link. I hope it will be useful.

Best regards

http://www.ifr.ac.uk/netsci08/Download/CT26_Toro_network/CT262_Leskovec.pdf

Thank you, Andrea. That is very helpful.

But what are other properties that allow to characterize communities or behaviour of "objects" in the communities?

As my friend said condactance is one of the community quality function, but the most famous quality function is modularity. This function can be used for assessing quality of found communities. Big problem of this function is resolution. For more studies you can refere to community detection in graph, very good survey written by Fortunato. 

But about other properties, from structural perspective we can define communities as dense inter connected nodes. But in other field other definitions exist. The most general definition of community is a group of similar nodes. You can specialize this definition for your field. 

Thank you, Hanif. I'll take a look at Fortunato's work.

Dear Olga, If you are talking about social communities, you may look at socially constructed identities in such communities or, in other words, symbolic entities more or less similarly perceived / shared by the members. One can be interested in shared understanding (or a degree of it) of community members about their community and interaction with the world. Sort of collective reflection on who we are. The issue of boundary would be interesting - how do community members distingush themselves from not members?