I never understood why the conservative party names themselves the Republicans, whereas the liberal party names themselves the Democrats. Yet, the US is a Republic, not a Democracy. Ironic...
It's an political artifact from the 1800s.
They were originally one party, the Democratic Republican party, founded by Thomas Jefferson and James Madison in 1792, in opposition to the Federalist party led by Alexander Hamilton.
They split up in the 1820s into the Jackson Men (led by Andrew Jackson, which became the Democratic party in 1844) and the National Republican party (Led by Henry Clay and John Quincy Adams, which was absorbed by the Whig party in 1832 after Clay was defeated in the presidential election, which then became the modern Republican party in 1854).