When people say "the government in Washington," they usually mean one of three distinct things:
Refers to the President, Vice President, Cabinet secretaries, White House staff, and the 2+ million civilian federal employees they oversee. Example: "The Biden Administration announced new EPA rules." It's a time-bound term—changes every 4–8 years.
Metonym for the U.S. Congress: the Senate (100) and House of Representatives (435). Named for the Capitol building where they meet. Example: "Capitol Hill is negotiating the defense bill." Represents lawmaking, oversight, and the "people's branch."
Literally the President's residence/office at 1600 Pennsylvania Ave. Figuratively, the nerve center of executive decision-making. Example: "The White House pushed for the infrastructure bill." Often used interchangeably with "The Administration" but carries more symbolic weight.
Bonus: "The Federal Government" = all three branches + independent agencies + judiciary. The umbrella term.