The Order of the Phoenix is introduced in the fifth book in the Harry Potter series, Harry Potter and the Order of the Phoenix, as a secret society of wizards and witches organized to fight against the power and influence of Lord Voldemort and his minions.
The word order has been used for centuries to refer to religious or other groups that have organized around a guiding principle or set of beliefs. It came into English from Old French, ordre, ultimately from Latin, ordo, meaning a row or series. We can see this meaning of the Latin source in words like ordinal (number) or ordinary.
The word phoenix refers, in classical mythology, to a rare bird that lived for five or six hundred years in the Arabian desert, and then burned itself on a funeral pyre, only to rise anew from the ashes and live again.
The word phoenix came to English from the Old French fenix, and ultimately from the ancient Greek phoenix, meaning Phoenician(s), or reddish purple, or of course the legendary bird itself. It’s also the name of the capital of Arizona, a constellation of stars west of Grus, and a group of islands that form a part of Kiribati.
The name Order of the Phoenix is not original to Rowling’s work. Greece instituted its own Order of the Phoenix (Τάγμα του Φοίνικος) in May, 1926, to recognize Greek citizens who have excelled in the arts and literature, science, public administration, shipping, commerce, and industry, and to foreigners who have helped raise Greece's international prestige.
Of course, Greece’s governmental award probably had nothing to do with Rowling’s choice of name for the organization that fought against Lord Voldemort. And the name certainly is appropriate, if foreboding for its members.