Flag Day marks the adoption of the Stars and Stripes by the Continental Congress on June 14, 1777. As explained in President Obama’s Proclamation of Flag Day and National Flag Week:
To commemorate the adoption of our flag, the Congress, by joint resolution approved August 3, 1949, as amended (63 Stat. 492), designated June 14 of each year as “Flag Day” and requested that the President issue an annual proclamation calling for its observance and for the display of the flag of the United States on all Federal Government buildings. The Congress also requested, by joint resolution approved June 9, 1966, as amended (80 Stat. 194), that the President annually issue a proclamation designating the week in which June 14 occurs as “National Flag Week” and call upon citizens of the United States to display the flag during that week.
The entire presidential proclamation is available on the White White House website, here.