Here are the official days to fly a flag at half-mast:
Peace Officers Memorial Day, May 15th
Memorial Day, last Monday in May
Korean War Veterans Day, July 27th
Patriot Day, September 11th
Pearl Harbor Remembrance Day, December 7th

Only the president of the United States or a state governor can order half-mast.  They determine the length of time too.  The length of time for other occasions is follows:
Thirty days after the death of a president or former president
Ten days after the death of a vice president, the chief of justice or a retired chief of justice or the speaker of the house of representatives.
Until the burial of an associate justice of the Supreme Court, secretary of a military department, a former vice president, or the governor of a state, territory, or possession.
On the day of and the day after the death of a member of Congress.
On Memorial Day the flag is flown at half-mast until noon, then raised to full-mast until sundown.
To fly a flag at half-mast, raise it to the top of the mast and lower it to the middle.  Before taking it down for the night, raise it to the top again.  Other flags on the pole should be removed or flown at half-mast also.

Picture is of flag at half-mast after President Lincoln’s funeral, courtesy of Library of Congress.