Presidents' Day; Presidents Day.
The spelling with the apostrophe is better and more common. Until 1971, Lincoln's Birthday (12 February) and Washington's Birthday (22 February) were both observed as federal holidays. In 1971, President Richard Nixon proclaimed that the two holidays would be combined into one, "the Presidents' Day," honoring all past presidents of the United States.
The singular possessive--President's Day--is clearly wrong but still occasionally appears.
Good to know. And, also, it's crying shame that Garner only has 239 Twitter followers. He is the best person in the whole world at knowing Modern American usage.