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.
The United States is a federal constitutional republic, in which the President of the United States (the head of state and head of government), Congress, and judiciary share powers reserved to the national government, and the federal government shares sovereignty with the state governments.n addition, the first President of the United States, George Washington, was not a member of any political party at the time of his election or throughout his tenure as president. Furthermore, he hoped that political parties would not be formed, fearing conflict and stagnation.