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.
Benjamin Harrison (August 20, 1833 - March 13, 1901) was the 23rd President of the United States, serving one term from 1889 to 1893. Harrison was born in North Bend, Ohio, and moved to Indianapolis, Indiana at the age of 21, where he became a prominent state politician.