Multiple members of the Cabinet deal with foreign relations, however Secretary of State Hillary Clinton specializes in the affairs of foreign nations
The Secretary of State.
The State Department, under direction of the President, deals with foreign policy.
Countries do not have affairs in the way you imply in your question. Countries have foreign relations with other countries and the department that deals with these relationships is called the foreign affairs department.A foreign affairs department will have relations with all countries. some of which will be ambassadorial relationships.
Most countries will offer foreign exchange. Countries like Australia and USA offer good deals. Although they accept students, some students from other countries may experience racism.
committee on foreign relations
(Committee on Foreign Relations)
It could make deals with other countries just like the United States.
The role that the president plays when he deals with foreign leaders is to negotiate on behalf of a nation. He acts as the representative of the entire nation.
The Department of State would be the US Department that deals with US foreign policy within the executive branch.
The Department of Interior
Secretary of Treasury
The department of firemen.