In the early days (George Washington, John Adams, Thomas Jefferson, etc.) there was no such thing as "Democrats" as a party. That came later.

There were the federalists and nationalists. Those words meant something different in those days than they do today The nationalists wanted a strong, central government. The federalists were opposed to that because they knew that history showed that centralized political power lead to disaster. They wanted decentralized power through a federalist system (the States would have the power and the federal government would be minimal).

Alexander Hamilton advocated for a king. He was a strong central government guy, and therefore a nationalist. But he started the "Federalists" faction (party) because he knew that his opinion was in the minority and he wanted to deceive the voters.

In response, the true federalists (lower case F) decided to call themselves the "Anti-Federalists." They wanted a REALLY small central government and were even opposed to the Constitution, as they prefered the framework of the Articles of Confederation (a true federalist document).

Washington and Adams were both nationalists and became part of the Federalist Party. However, they were the only two presidents who won under that ideology.

In response to all of this, Jefferson called his party the "Republican Party." But this was not today's Republican Party. Jefferson was in favor of a republican form of government, which decentalizes power. The next several presidents who won were all Republicans of this type.

The Republicans had been so successful, that there was no room for other ideas. That is when the Democrat Party was created, and Andrew Jackson was their first president. This is the same Democrat Party as today. Their first president was a slave owner. LOL.

Jefferson's Republican Party eventually split. Some went with the Democrats, others created the Whigs and the Anti-Masons, and other small parties.

In the 1850's, the current Republican Party was created, with Abraham Lincoln becoming their first president. Though they had the same name, they did not have the same ideology as Jefferson's Republican Party.

This is why today the mainstream (read: mostly false) historians call Jefferson's Party the "Democrat-Republican" Party. But that is not what Jefferson and his people called themselves.