The Alliance refers to the alliance formed between the Explorers, the Cygnus Knights, the Resistance and the Heroes formed by Cygnus to prepare for the oncoming threat of the Black Mage and the Black Wings. The alliance was introduced during the Advance of the Union update.
Upon achieving Level 70, accept Cygnus's quest to be warped to Ereve. Talk to Cygnus and she will give you the above mentioned medal which gives HP +500 and MP +500.
The Alliance takes action directly several times in events like Black Heaven, also in the quest line in Twilight Perion, where everyone has the same dark dream and Cygnus sends the player to investigate the situation through the Gate to the Future.