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.
In the context of the debates over the ratification of the Constitution, Thomas Jefferson was neither a staunch Federalist nor an Anti-Federalist, but rather an ambivalent Federalist who thought the Constitution needed to have a Bill of Rights add...