It is to be distinguished from the Council of Europe, which is a completely seperate international organization, and the European Council, which is a seperate (though closely related) EU institution.
The Council is assisted by COREPER, which consists of representatives of the member-state at the level of diplomats or high-level civil servants. COREPER generally prepares the Council agenda, and negotiates minor and non-controversial matters, leaving controversial issues for discussion by the Council.
While legally speaking the Council is a single entity, it is in practice divided into several different councils, each dealing with a different functional area. The General Affairs Council is the most important of these, containing Foreign Ministers. Each council contains a different type of minister: e.g. the Agriculture Council contains Agriculture Ministers, and so on.