These are largely just different terms for the same things, but named differently to remind you of some important details. For a rectangle, we usually refer to the length and width; properly, these are the longer side and the shorter side, respectively, but that really makes no difference.
John Adams hated political parties, but he was backed by the Federalists and ran for President as a Federalist and won.   After Adams served, Thomas Jefferson, James Madison James Monroe served two terms each as members of the Democratic Republican party. By this time the Federalist Party all but disappeared.
Grover Cleveland. Grover Cleveland was the only Democrat to serve as president in the second half of the 19th century, and he was arguably the last conservative Democratic president in U.S. history.