Jefferson and others formed the Democratic-Republican Party to oppose the Federalists.
After the 1824 election, the party split into the Democratic Party (centered around Andrew Jackson) and the National Republican Party. The National Republican Party dissolved after the 1832 election.
Since the Democratic Party is still around, it is more accurate to say Jefferson founded the Democratic Party, not the Republican Party.
In 1833, the Whig party formed from some of the ashes of the National Republican Party, along with some Federalists (Jefferson opposed Federalist, remember), and some people who had fallen out with the Jacksonian Democrats.
The Whigs managed to get two guys elected President, both of whom died in office. Son of a...*****!
The Whigs imploded over the question of the expansion of slavery. From its anti-slavery ashes arose the modern day Republican Party in...1854.
Welcome to your first REAL lesson in American History.