Washington D. C. is the nation's capitol.  It is neither a state or the part of one.  Rather it is a district.  It was proposed by an act of congress in 1790, established in 1791, and the first session of congress was held in 1800.  John Adams was the first president to live in the White House.
All three branches of the federal government reside here.  The legislature, the house and senate, are located on Capitol Hill.  The judiciary resides in the  Supreme Court building.  The executive branch, led by the president, is housed in the White House.

In addition to a multitude of government buildings and federal agencies many monuments and memorials are in and around the D. C. area.  Also in close proximity is the Arlington National Cemetery,  the Washington Cathedral, The Pentagon, the Verizon Center (home to the Capitals of the NHL and the Wizards of the NBA), Chinatown, Embassy Row, Ford's Theater, the African-American Civil War Museum, the Smithsonian's, the Holocaust Museum, the Spy Museum, and National Park (home of the baseball Nationals).  Most of these venues are accessible via the Metro.