What branch decides to recognize foreign governments?

Recognition of foreign governments is normally shown by the establishment of a consulate, embassy, or special mission to the foreign government in question. These embassies (etc) are established and run by the US Department of State, headed by the Secretary of State (currently Hillary Clinton) and overseen by the President. Therefore, recognition of foreign governments is typically done by the executive branch.

Congress has the power to ratify treaties with foreign governments and can pass legislation involving the recognition of or relationships with foreign governments. So sometimes, recognition of foreign governments can be done by the legislative branch.

But generally speaking, recognizing foreign governments is a power of the Head of State of a nation, and the Head of State of the United States is the President, so it's an executive branch power.