Flag Day commemorates the adoption of the flag of the United States (wiki) which happened on June 14, 1777 by resolution of the Second Continental Congress:
That the flag of the United States shall be of thirteen stripes of alternate red and white, with a union of thirteen stars of white in a blue field, representing the new constellation.
The resolution was made following the report of a special committee which had been assigned to suggest the flag’s design.
A flag of this design was first carried into battle on September 11, 1777, in the Battle of the Brandywine. The American flag was first saluted by foreign naval vessels on February 14, 1778, when the Ranger, bearing the Stars and Stripes and under the command of Captain Paul Jones, arrived in a French port. The flag first flew over a foreign territory in early 1778 at Nassau, Bahama Islands, where Americans captured a British fort.
Lest you think it's lost its power, remember what the flag can still accomplish:
Two years earlier, on June 14, 1775, Congress adopted "the American continental army", so today is also the Birthday of the U.S. Army. More detail here, at the Army's web site.