Does Washington D.C. have a governor?
The term "governor," in American politics, refers to the head of a state. "Washington" is another name for the District of Columbia; a "district" is basically equivalent to a city, so its administrative leader is a mayor.
Washington, D.C., has been organized and governed differently throughout its history. When it was first created in 1790, the District of Columbia was governed by a three-person council appointed by George Washington: Thomas Johnson, Daniel Carroll, and Dr. David Stuart. The first leader to hold the position of mayor was appointed in 1802.
For a summary of Washington D.C.'s development, try the history on the Washington, D.C. home page.