The House and Senate Committees
The Senate committee has a special role to play because the Senate has to approve all treaties and all nominations to key foreign policy postings. Members, therefore, have a great deal of influence on how U.S. foreign policy is conducted and who represents the United States around the world.
The House committee has less authority, but still plays an important in passing the foreign affairs budget and in investigating how that money is used. Senate and House members often travel abroad on fact-finding missions to places deemed vital to U.S. national interests.
Certainly the most important authority given to Congress overall is the power to declare war. The authority is granted in Article 1, Section 8, Clause 11 of the U.S. Constitution. But there has always been a tension between this and president's constitutional role as commander-in-chief of the armed forces. In 1973, Congress passed the controversial War Powers Act to clarify how the president could carry out military action while still keeping Congress in the loop.
Congress, more than any other part of the federal government, is the place where special interests seek to have their issues adressed. And this creates a large lobbying and policy crafting industry, much of which is focused on foreign affairs. Americans concerned about Cuba, agricultural imports, human rights, global climate change, immigration, and much, much, more seek out members of the House and Senate to influence legislation and budget decisions.