A variety of organizations exist in the Wizarding World. Some of them are official, others were formed by various witches and wizards to promote a particular cause.
Quidditch league that includes teams from Australia and New Zealand (QA8)
The Brotherhood of Goblins
A group pressing for goblin rights, including the right to carry and use wands. Members of the B.O.G. were meeting with Ministry representatives in Chipping Sodbury when a small riot broke out. Some goblins apparently got hold of real wands and were transfiguring post boxes and litter bins into wildebeest and marching through the streets chanting B.O.G. slogans. The leader of the organization is Brodrig the Boss-Eyed (DP)
Campaign for Greater Freedom for Wizards
Glinda Crook spoke for this group in the Daily Prophet, decrying the actions of the Ministry of Magic designed to reign in the actions of wizard-kind on Hallowe'en (DP).
More commonly referred to as the D.A. (see), the name of the organization was chosen by Ginny Weasley to go with the initials (suggested by Cho Chang as an acronym for another possible name). The name "Dumbledore's Army" was selected because it signified the worst fear of certain high-ranking officials in the Ministry of Magic at the time.
Fresh Air Refreshes Totally
Dedicated to wearing robes instead of trousers, even around Muggles. The president of F.A.R.T. is Archie Aymslowe (Pm: Wizard Clothing).
Goblin Rights groups
Over the past centuries, various groups have advocated goblin rights, sometimes with violence. In the 1990s, some goblin groups have been meeting with representatives of the Ministry of Magic, hoping to create a Bill of Goblin Rights (DP). Oswald Beamish was a pioneer of goblin rights (fw).
Order of Merlin
Although we now think of the Order of Merlin as an award, it was originally conceived by Merlin himself as an organization which made rules against using magic on Muggles (fw).
Society for the Promotion of Elvish Welfare
Founded in 1994 by Hermione Granger in response to what she saw as gross injustice in the treatment of house-elves. The Society didn't exactly catch on among Hogwarts students, but she persevered. Eventually (fall of 1995) she started knitting hats and socks which she left lying around Gryffindor tower, hoping to free some unsuspecting elf who picked them up while cleaning the common room. Once the elves discovered the hidden garments, they were deeply offended and refused to clean Gryffindor Tower anymore. Dobby picked up the slack, doing all the cleaning himself (GF, OP).
Society for the Support of Squibs
Founded by Idris Oakby in the early-to-mid 1900s (JKR).
Society for the Tolerance of Vampires (S.T.V.)
This society has a candle-lit office in London. They advertised for someone to run the office, with preference to be given to applicants with a garlic allergy (DP).
Troll Rights Movement
The spokesperson for this movement is Heliotrope Willis. She speaks optimistically about trolls' essentially peace-loving nature and of the valuable services they provide to wizards by providing protection (DP)