Quote from wikipedia:
...the [Republican] Party is not to be confused with the Democratic-Republican party of Thomas Jefferson or the National Republican Party of Henry Clay...However, During Jefferson's presidency, he was called a "Republican", but the reference was to the party now known as the Democratic-Republican Party. That party later split into the Democratic Party and the Whig Party. The latter was formed in the winter of 1833-1834 but was defunct by the time of the American Civil War.
Interesting explanation of how the confusion can occur.

Did you know the American political system is designed (intentionally or unintentionally is up for debate) to function effectively as a two-party system?

A third (or even a fourth) party can and will exist from time to time but eventually one or more parties will fail or lose any significant clout until only two parties remain on the national level. State politics work a little differently allowing for multiple parties.