The Republican and Democratic parties, which are the main parties in the United States, derive from the same party, the Republican Democrats that derive from the Federalist Party.
The Federalist Party was founded around the US Constitution. Once approved, and in the midst of the controversy over whether the federal government should govern all aspects of national life, or if states should have autonomy, some federalists separated from the party and founded the Republican Democrats, who believed in a autonomy of the states.
The Federalist Party that had been that of George Washington, and other founding fathers, was extinguished in 1816.
From there there was another separation, based on the concept of slavery, was when supporters and detractors of slavery were divided, around the presidential election in 1824, when John Quincy Adams was elected. The pro-slavery splintered to form an independent party: the Democratic Party and the others the Whig Party, to which Abraham Lincoln belonged.