Federation Day

Federation Day is a day of celebration commemorating the founding of the United Federation of Planets.

A card game called "Federation Day" poker existed in the 24th century, in which "2"s, "aces", and "6"s were wild, since the Federation was founded in 2161.

Deanna Troi suggested playing this game to the rest of the senior staff members of the USS Enterprise (NCC-1701-D) in 2368, however, Worf found it difficult to ascertain exactly what he had in his hand with so many wild cards. 

Background
Non-canon sources have established several dates for the event:
 * October 11 - "It's Federation Day!", a newspaper clipping created for but not seen on screen.
 * August 12 - in the last chapter of the novel Last Full Measure
 * May 8 - background in the Star Trek: Star Charts
 * June 30 - Listed on a memorial plaque at Starfleet Academy in Star Trek Online.