Armagest serves as the primary ghost/villain organization of Kaze no Stigma. This european magic organization is referred to as "The Order of Magical Arts," and is known to be the ultimate authority when it comes to their unparalleled skill in modern magic. From the four members that were introduced in the series: († = Deceased).
It's at least safe to assume that from what Erwin did to Tsui Ling, how Michael manipulated Misao and the lengths Bernhardt went to create the "game" known as Pandemonium, Armagest plans to gain power by sacrificing the innocent to powerful demons to aid their cause in taking over the world. As a side objective, Armagest has held a deep grudge against Kazuma for killing its previous leader, Erwin Leszar after he'd recently sacrificed Tsui Ling to a powerful demon.
From then on, Armagest has made plenty attempts to avenge Erwin, first by sending his former subordinate Michael Harley to deal with Kazuma and his group accordingly. Then after that attempt failed, Bernhardt sought to bring his nemesis the ultimate despair by creating a clone of Tsui Ling (Lapis) as his personal subordinate and to create a game known as Pandemonium.
Although the game initially made it seem like normal people were becoming magic users, Bernhardt was actually implanting yoma into the players, where they would become hosts (or seeds in the anime). On a certain day when the clock stroke midnight, the yoma would emerge from the hosts and transform them into complete monsters -only for Kazuma to slay them all and the slain creatures would then be sacrficed to summon the King of Yoma just long enough for it to destroy Tokyo and bring Kazuma and his comrades the ultimate despair.
With all of Bernhardt's plans and attempts for revenge foiled, both he and Lapis had no other choice but to flee. However, it wasn't before Lapis told Kazuma what Tsui Ling's final thoughts were before they left..
Armagest (or Almagest) is a universal term given to any of the celebrated textbooks on astrology and alchemy used during the Middle Ages.
The term Almagest also translates as "The Greatest."