"Third party" political parties have traditionally been parties with limited platforms such as the Anti-Masonic Party or the Prohibition Party, ones with specific philosophical bases such as various socialist parties, or those coalescing around specific personalities, such as the Progressive (or "Bull Moose") Party centering on former President Theodore Roosevelt. "Third party" is somewhat of a misnomer, as frequently there are more than one operating at a time. Such parties contested the presidential election of 1832 and every one since 1840.
Some of these have been splits from the two major parties (the Democratic-Republican Party which became the Democrats, the other the Federalists then Whigs then Republicans). Splits have produced parties such as the Free Soil Party and the Progressives. Other third parties are complete alternatives to the major parties, including the Socialist, Socialist Workers and Communist Parties, as well as America First, Populist, American Independant, etc. Some have represented specific programs, and others merely discontent. Many of these parties have achieved succes at the local, state and federal levels, especially the Libertarians, although there has been only one third-party president. This was Lincoln in his re-election campaign, running with Democrat Andrew Johnson as his V.P. candidate on the New Republic ticket in 1864. This is usually overlooked, with Lincoln's re-election referred to as Republican with no explanation as to why his V.P. was a Democrat.
Many third-party candidates have been influential and popular figures, such as T. Roosevelt, who took the anti-Big Business Republicans with him to the Progressive Party in 1912, Douglas MacArthur's 1952 campaign and Ralph Nader's campaigns. Without the funds and party machinery of the major parties they have remained unsuccessful. Some "third-party" candidates have actually represented more than one political party in the same election, as MacArthur with the Constitution and America First Parties and Robert LaFollette's 1924 campaign as candidate for both the Progressive and Socialist Parties.
Some of these groups have influenced the major parties. The Prohibition Party eventually mustered enough support to get the 18th Amendment passed in Congress. Various platform planks of some of these parties have been co-opted by the major parties, such as the central bank and graduated income tax planks of the Communist Party. In 1872 Victoria Woodhull of the Equal Rights Party became the first woman to win a party nomination for president, with Cynthia McKinney of the Green Party in 2008 the most recent. Third party candidates have run the gamut from George Wallace (American Independant), Dick Gregory (Freedom and Peace) and Eldridge Cleaver (Peace and Freedom), all in 1968, to left-winger turned right-winger Lyndon LaRouche, from Strom Thurmond's 1948 run for the States' Rights Party to Benjamin Spock for the People's Party in 1972.
Occassionally third-party candidates have split enough votes from one of the major parties to tilt the election toward the other major party, as in Ross Perot's Independant campaign of 1992 and Ralph Nader's unfortunate 2000 campaign. There have been at least 39 "third parties" in presidential elections and many more in local and state elections. The third link below leads to a list of all (except the New Republic) such parties and their candidates in US presidential elections.