What is Washington, DC?

DC merely stands for the District of Columbia.

What exactly does the District of Columbia mean?
Regarding its meaning, Columbia became a popular nickname for the North American British Colonies and then the United States of America during the 18th and 19th centuries. Poetry, books, speeches, and songs all referred to the United States as Columbia. So, the byname of District of Columbia really just means "District of the United States".

Why is Washington, DC, referred to as the District of Columbia?
Beginning in the early 1800s, the US Congress began referring to the territory officially as the "District of Columbia" in legislation that it passed. Many towns and counties across the new nation were already named Washington by the time Congress authorized the creation of the federal city along the Potomac River in 1790. Using the byname of "District of Columbia" distinguishes the city from others called Washington for practical purposes.

Why do they call the city a district?
A federal district in the United States is distinct from a US federal territory such Puerto Rico or the US Virgin Islands. The city is called a district, because its area falls under jurisdiction of the federal government as opposed to state government. Washington is officially a district that is a federally administered area with limited self-government.

How did the city of Washington, DC, get its name?
In 1790, the US Congress passed the Residence Act, which stipulated that an area along the Potomac River be the location of the country's new capital. A year later, three commissioners were given the responsibility of preparing and purchasing for the US Government land that President George Washington had chosen as the specific site for new capital. That year, they named the federally-controlled area the "Territory of Columbia" while the incorporated section within it (i.e., the smaller section with commercial and government buildings) was named "City of Washington"--in honor of George Washington. Early on, the city covered only a small fraction of what we today call the district. By 1801, the US Congress referred to the area as the "District of Columbia" in legislation.