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A running mate is a person running together with another person on a joint ticket during an election. The term is most often used in reference to the person in the subordinate position (such as the Vice President running with a presidential candidate) but can also properly be used when referring to both candidates, such as "Michael Dukakis and Lloyd Bentsen were running mates in 1988".
The term is usually used in the United States, in reference to a prospective Vice President. In some states, candidates for lieutenant governor run on a ticket with gubernatorial candidates, and are also known as running mates.
In United States politics 
In the United States, "running mate" refers to a candidate for vice president (federal) or, in those states where the governor and lieutenant governor are elected together, to a candidate for lieutenant governor (state). Historically, running mates were chosen by political parties in consultation with the principal candidate (i.e., the person running for president or governor). In the late 1960s, it became the practice of the principal candidate in presidential elections to announce his preferred choice of running mate at his political party's national convention. The current practice is for the presumptive nominee of a political party to announce his choice for running mate before the national convention which, because of the extensive primary election and caucus system, is becoming increasingly irrelevant.
Some states (e.g., Michigan) exactly mirror the federal process above for their gubernatorial elections, while others continue to hold separate votes for gubernatorial and lieutenant gubernatorial candidates; and some states have abolished or never had the office of lieutenant governor.
Meanwhile, some states, like California and Texas, still elect the governor and lieutenant governor on separate tickets, and frequently from different political parties. This was the case when George W. Bush was Governor of Texas. His first lieutenant governor, Bob Bullock, was a Democrat. In cases like this, the governor and lieutenant governor are not considered running mates because they are not elected on the same ticket.
The practice of running candidates for president and vice president together evolved in the nineteenth century. Originally, electors cast two ballots for president and whoever took second place in the tabulation became vice president. Starting in 1804, the president and vice president were elected on separate ballots. As more and more states subsequently began to choose their electors by popular election instead of appointment (South Carolina being the last state to change, in 1860), candidates began to realize they could run together as a team for president and vice president instead of running completely separately for each office.
The practice of a presidential candidate having a running mate was solidified during the American Civil War. In 1864, in the interest of fostering national unity, Abraham Lincoln from the Republican Party (popular in the North) and Andrew Johnson of the Democratic Party (popular in the South), were co-endorsed and run together for President and Vice-President as candidates of the National Union Party. Notwithstanding this party disbanded after the war ended, with the result that Republican Lincoln was succeeded by Democrat Johnson; the states began to place candidates for President and Vice-President together on the same ballot ticket - thus making it impossible to vote for a presidential candidate from one party and a vice-presidential candidate from another party, as had previously been possible.
Running mates are typically chosen to balance the ticket geographically, ideologically, or personally, as when the staid New Englander John F. Kennedy was matched with the folksy Texan Lyndon B. Johnson in 1960. Or, in 1984, when male Walter Mondale was paired with female Geraldine Ferraro. The object is to create a more widespread appeal for the ticket.
Presidential candidates from smaller states sometimes choose a vice presidential running mate from a state with a large number of electoral votes - as when Walter Mondale of Minnesota (10 votes) selected Geraldine Ferraro of New York (then 36 votes). It is preferred, but not legally required, that the running mate be from a different state from the presidential nominee, because each elector can vote for no more than one candidate from his or her own state. Running mates can also be chosen from swing states, for example Mitt Romney nearly picked Senator Rob Portman from Ohio in order to boost his chance of winning in the state.
In electing a subordinate officer the Electors will not require those qualifications requisite for supreme command. The office of the Vice President will be sinecure. It will be brought to market and exposed to sale to procure votes for the President.