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.
The election of 1912 was between four candidates. William Howard Taft was the incumbent at the time and was nominated by the Republican Party to represent the conservative wing. Theodore Roosevelt also sought nomination from the Republicans, but failing to receive it, he chose to run with the Progressive ("Bull Moose") Party. Woodrow Wilson ran for the Democratic Party, and Eugene V. Debs for the Socialist Party. Wilson swept away the competition and won the election easily.
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).
Yes, an incumbent president has lost his party's nomination before. One notable example is President Jimmy Carter, who lost the Democratic Party's nomination to challenger Ted Kennedy in 1980.
Yes, there have been instances where an incumbent president has lost the nomination for their party. One notable example is President Franklin Pierce, who failed to secure the Democratic Party's nomination for a second term in 1856.