United States has a First Past the Post (FPTP) electoral system.
US Constitution? Specific individual states' rules? Party rules?
Related to this question.
To make it less broad, US Presidential election is the main scope of the question.
Elector college representative's voting procedures are defined by US Code title 3 chapter 1 section 8, which states
Manner of voting
§ 8. The electors shall vote for President and Vice President, respectively, in the manner directed by the Constitution.
Amendment XII of the constitution states
The Electors shall meet in their respective states, and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and all persons voted for as Vice-President and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate.
The President of the Senate shall, in the presence of the Senate and House of Representatives, open all the certificates and the votes shall then be counted.
The person having the greatest Number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. But in choosing the President, the votes shall be taken by states, the representation from each state having one vote; a quorum for this purpose shall consist of a member or members from two-thirds of the states, and a majority of all the states shall be necessary to a choice. And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following, then the Vice-President shall act as President, as in the case of the death or other constitutional disability of the President.[Note 1]
The person having the greatest number of votes as Vice-President, shall be the Vice-President, if such number be a majority of the whole number of Electors appointed, and if no person have a majority, then from the two highest numbers on the list, the Senate shall choose the Vice-President; a quorum for the purpose shall consist of two-thirds of the whole number of Senators, and a majority of the whole number shall be necessary to a choice. But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
In other words, if someone manages to get a majority, they win. If they don't there is a run-off-ish election in the congress and senate for President/VP respectfully. (Side Note: If you heard the story about Evan McMullin potentially becoming president after only winning one state, this is how that would have happened). This is technically a majority rules election, not a FPTP election (Thanks Brythan).
Note that this is only for the second, direct presidential election by the electoral college representatives. The more known general election (IE what we all did on Nov 8th) which, in most cases, binds the electoral college representatives to their state's popular vote have different laws governing them that vary state to state. I randomly picked Delaware as an example:
§ 4303 Meeting and voting of electors.
(a) The electors chosen or appointed in this State for the election of a President and Vice-President of the United States shall meet and give their votes at Dover on the day determined by Congress for that purpose.
(b) In all cases, the electors chosen or appointed in this State for the election of a President and Vice-President of the United States under this chapter shall be required to cast their individual votes in accordance with the plurality vote of the voters in this State.
