The U.S. Congress passes bills that become federal laws when the U.S. President signs them (although there are cases in which Congress can make a bill into a law without the President's approval), and each state Congress passes bills that become state laws when the Governor of the state signs them.
There are two heads of the US Senate; the Senate Majority Leader and the US Vice-President. The US Vice-President, who is President of the Senate, only breaks tie votes.
John Quincy Adams
The President can veto bills passed by Congress, thus checking the Legislative Branch. The President checks the judicial branch by appointing Supreme Court justices.
The primary elections are for delegates to the National nomination convention. Not all states have them and the way in which they select delegates varies. Sometimes the delegation is divided in proportion to the vote, sometimes the one with the most votes get all of the delegates. Sometimes the primaries are only advisory information for the delegates. If no candidate has a majority of the bound and committed delegates by convention time, delegates are all free after the first ballot to vote as they choose. Ballots are taken and deals are made and delegates switch votes until finally one candidate has a majority.