"The executive Power shall be vested in a President of the United States of America. He shall... be elected, as follows: Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress..."
The U.S. Electoral College was established in Article II, Section I of the United States Constitution and was later modified by the Twelfth and Twenty-Third amendments, which clarified the process. The term "Electoral College" does not actually appear in the Constitution, but instead, refers to the process of how the President is to be elected in the United States. (Source: NARA )
Regarding this process, Founding Father James Wilson said, “This subject has greatly divided the House, and will also divide people out of doors. It is in truth the most difficult of all on which we have had to decide.” (Source: Constitution Annotated)
While the Electoral College was established in the Constitution, the details of the process are governed by Chapter 1 of Title 3, United States Code. This Chapter on Presidential Elections and Vacancies includes provisions on the appointment of electors, Certificates of Ascertainment and Vote, counting votes in Congress (see below), and more.
Chapter 1 of Title 3 also requires the Archivist of the United States to perform several duties that have been delegated to the Director of the Office of the Federal Register. These duties include coordinating between the States and Congress as well as making the physical Electoral College Certificates open to public inspection for a year after the election. For a more in-depth look into the Electoral College and how it works, visit the National Archives.
Section 15 of 3 U.S.C. Ch. 1 requires the Senate and House of Representatives to meet on the sixth day of January following every meeting of the electors to verify the certificates and count the votes of the electors. Read the proceedings in the Congressional Record back to 1881: