The Council of the European Union is an Institution with a formal legislative role. It is composed of different configurations of national Ministers (one per country). The exact configuration of each meeting depends upon the topic under discussion.
The ECOFIN Council is a monthly meeting, composed of the Economics and Finance Ministers of the Member States, as well as Budget Ministers when budgetary issues are discussed. Ireland is represented by Minister for Finance, Michael Noonan TD, and Minister for Public Expenditure and Reform, Brendan Howlin TD.
For more information see: http://www.eu2013.ie/events/event-items/meetingofeconomicandfinancialaffairscouncilecofin-20130621/