Federation Day
Federation Day was 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  in 2368, however, Worf found it difficult to ascertain exactly what he had in his hand with so many wild cards. ()

In 2161, after the charter was signed, ratifying the Coalition of Planets, which ultimately lead to the formation of the United Federation of Planets. ()

Background information 
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  Last Full Measure, To Brave the Storm, and . 
 May 8 – background in the Star Trek: Star Charts.
 June 30 – Listed on a memorial plaque at Starfleet Academy in Star Trek Online.
Celebrations
Federation