I didn't remember when I was asked the other day, so I looked up the history of Flag Day. From Wikipedia:
In the United States Flag Day is celebrated on June 14. It commemorates the adoption of the flag of the United States, which happened that day by resolution of the Second Continental Congress in 1777. The United States Army also celebrates the Army Birthday on this date; Congress adopted "the American continental army" after reaching a consensus position in the Committee of the Whole on June 14, 1775.
In 1916, President Woodrow Wilson issued a proclamation that officially established June 14 as Flag Day; in August 1949, National Flag Day was established by an Act of Congress.