Every party that has had incumbent presidents in office has renominated at least one of them except the Whig Party.
The party who does not have a 'the' candidate in office, incumbent means that that parties candidate already holds the office.
The incumbent, Gerald R. Ford, ran for another term in 1976. Bob Dole was his running mate.
Democratic Party candidate Woodrow Wilson won the 1912 presidential election defeating Progressive Party (nicknamed the "Bull Moose Party") candidate Theodore Roosevelt and Republican Party candidate incumbent President William Taft.
Former Republican President Theodore Roosevelt ran as a third party candidate against incumbent Republican President William H. Taft in 1912. Democrat Woodrow Wilson beat both of them (as should be expected when the votes on one side are split).
Ronald Reagan won the 1980 presidential election defeating incumbent President Jimmy Carter. Incumbent President Ronald Reagan won reelection in the 1984 presidential election defeating Walter Mondale.
If the incumbent president wants to run again for president, it ultimately depends on the rules and processes within each political party. Both the Democrats and Republicans typically hold primary elections or elect delegates to decide their presidential nominees. The decision ultimately rests with the party members and voters, who will choose whether to support the incumbent president or opt for a different candidate.
In the 1912 election, the most conservative candidate was William Howard Taft, who was the incumbent president and ran as the nominee of the Republican Party. However, Taft's conservative faction split from the main Republican Party, and a new party called the Progressive Party, also known as the Bull Moose Party, was formed. The Progressive Party nominated former president Theodore Roosevelt as their candidate, making him the most conservative candidate in the election.
In the U. S. Presidential Election of 1912, Progressive Party candidate Theodore Roosevelt received 88 votes while Republican Party candidate and incumbent President William Howard Taft received eight votes. Democratic Party candidate Woodrow Wilson won with 435 votes.
If an incumbent president is willing and qualified to run for reelection, his party does not contest his nomination, most likely because of the message that would be interpreted if the party does not support its incumbent president. However, that was not always the case. Before the Civil War, several incumbent presidents, including Franklin Pierce in 1856, sought but did not receive their parties' nominations for president.
In mid-2012, Romney is seeking the nomination by the Republican party as its Presidential candidate in the November, 2012 election (likely running against incumbent president Barack Obama).
The Democratic Party candidate for the 1904 presidential election was Alton B. Parker. Parker lost to the incumbent Republican candidate, Teddy Roosevelt.
They run as a team and are both elected by the same electors who have pledged to support their candidacy. The only way they could not be from the same party would be if no candidate received a majority of the electoral vote so the House of Representatives chose the President and the Senate chose the Vice-President.