The Six(ish) U.S. Party Systems

The First Party System (1780s-1815): Federalists and Democratic-Republicans
This system developed as a result of the factions in the Washington Administration. The two factions were Alexander Hamilton and the Federalists and Thomas Jefferson and the Democratic-Republican Party. The Federalists argued for a strong national government with a national bank and a strong economic and industrial system. The Democratic-Republicans argued for a limited government with an emphasis on farmers and states' rights. After the Election of 1800 the Democratic-Republicans gained major dominance for the next twenty years, and the Federalists slowly died off.
The Second Party System (1828-1860): National-Republicans/Whigs and Democrats
This system developed as a result of the one party rule of the Democratic-Republicans. The party was unable to deal with some of the most pressing issues of the time, primarily slavery and eventually split as a result of the "Corrupt Bargain" and the subsequent election of Jackson four years later in 1828. As the Democratic-Republican party split in 1828, those who supported John Quincy Adams formed the National-Republican Party while supporters of Jackson became the Democrats. In 1833 the National-Republican Party joined with the small Anti-Masonic Partyand formed the Whig Party in support of Henry Clay. Wealthier people tended to support the Whigs and Clay's American System. Poorer Americans tended to support the Democrats. Over time the Whig party began to break into factions over the issue of slavery and was supplanted by the Republican Party.
The Third Party System (1860-1896): Republicans and Democrats
The Third Party System began with the election of Lincoln on the Republican ticket in 1860. The Republicans were a relatively new party (formed in 1854) with the primary agenda of stopping slavery's expansion (though not necessarily ending slavery). This system was defined by bitter conflict and striking party differences and coalitions. The sectional conflict in the U.S. in the lead up to the Civil War caused these coalitions to be defined most evidently by geography. The South was dominated by the Democrats who opposed the ending of slavery, and the North, with the exception of some major political machines in the cities (most notably Tammany Hall in New York City), was dominated by the Republicans, who eventually came to support ending slavery. After the Civil War ended in 1865, this era became a time of extreme industrial and economic expansion. Other than Andrew Johnson (1865-1869) and Grover Cleveland (1885-1889 and 1893-1897), the Republicans held the presidency through this period.
The Fourth Party System (1896-1933): Republicans and Democrats
This period generally covers the Progressive Era and the Roaring Twenties. The period featured a transformation from the issues of the Third Party System, which had focused on the Civil War, Reconstruction, race and monetary issues. The era began in the severe depression of 1893 and the extraordinarily intense Election of 1896 (William McKinley (R) defeated William Jennings Bryan (D). The central domestic issues concerned government regulation of railroads and large corporations ("trusts"), the money issue (gold versus silver), the protective tariff, the role of labor unions, child labor, the need for a new banking system, corruption in party politics, primary elections, direct election of senators, racial segregation, efficiency in government, women's suffrage, and control of immigration. Foreign policy centered on the 1898 Spanish-American War, imperialism, the Mexican Revolution, World War I, and the creation of the League of Nations. The business communities of the Northeast generally supported the Republicans while the South and West supported the Democrats. Political machines controlled by both parties worked hard to ensure that they controlled the immigrant vote. With the exception of the eight years under Woodrow Wilson from 1915-1923, this period was dominated by Republicans.
The Fifth Party System (1933-1968): Republicans and Democrats
This system was defined by the creation of the New Deal Coalition by Democratic President Franklin D. Roosevelt in response to the Great Depression. This coalition supported new social welfare programs brought together many under-privileged, working class, and minority groups including unions, Catholics, and Jews. It also attracted African-Americans, who had previously largely supported the Republican Party due to Lincoln's freeing of the slaves. This era lasted until the election of Nixon in 1968. Civil Rights issues began to divide the Democrats along geographic lines in the 1960s and many Southern Democrats joined the Republicans in opposition to Civil Rights legislation pursued by John F. Kennedy and Lyndon B. Johnson. The only Republican president in this period was Dwight Eisenhower.
The Sixth Party System (1968-?): Republicans and Democrats
As the Democrats lost control of the South because of Civil Rights issues the Sixth Party System began. Though no party has been able to completely dominate politics, the Republicans have won the majority of the elections in the time period. The presidents were as follows:
Richard Nixon (Republican): 1969-1974
Gerald Ford (Republican): 1974-1977
Jimmy Carter (Democrat): 1977-1981
Ronald Reagan (Republican): 1981-1989
George H. W. Bush (Republican): 1989-1993
Bill Clinton (Democrat): 1993-2001
George W. Bush (Republican): 2001-2009
Barack Obama (Democrat): 2009-present