The Federalist Party and the Democratic-Republican Party were two important groups in early America. They had different ideas about how the country should be run.
First, on foreign policy, the Federalists, like Alexander Hamilton, wanted to be friends with Britain and trade with them. They thought it was important for America to be strong and rich. On the other hand, the Democratic-Republicans, led by Thomas Jefferson, liked France more. They wanted to support France because they helped America in the Revolutionary War.
Second, about the economy, Federalists believed in a strong national bank and wanted the government to help businesses grow. Hamilton thought this would make the country wealthy. In contrast, Jefferson and the Democratic-Republicans wanted to keep farmers happy and thought the common people should be in charge of the economy, not big businesses.
Finally, when it came to the federal government, the Federalists wanted a strong central government that could make big decisions for the whole country. They thought this would keep everything in order. But Jefferson and the Democratic-Republicans thought the states should have more power and that the federal government should be smaller so that the people could have more say in what happened.
So, in summary, Federalists liked Britain and a strong government, while Democratic-Republicans liked France and believed in giving more power to the states. They also had different ideas about money and business.