"D.C." stands for District of Columbia, the federal district created to serve as the United States' permanent capital. Here's the historical sequence:
Article I, Section 8 grants Congress power to establish a district (not exceeding 10 miles square) as seat of government.
Compromise between Hamilton (North) and Jefferson/Madison (South): capital on Potomac. Maryland and Virginia ceded land (100 sq mi total).
Commissioners named the district "Territory of Columbia" and the city "Washington" (after President Washington). "Columbia" was a poetic female personification of America, derived from Christopher Columbus.
Formally organized the District of Columbia; placed under Congressional control. Alexandria (VA portion) retroceded in 1846.
District of Columbia Organic Act merged Washington City, Georgetown, Washington County into single municipal government "District of Columbia."
"DC" is the postal abbreviation (like state codes). Residents lack voting representation in Congress (23rd Amendment gave Electoral College votes). Statehood movement seeks admission as "Washington, Douglass Commonwealth."
The name honors Columbus indirectly via "Columbia"—the feminine allegorical name for the New World—reflecting late 18th-century republican symbolism.