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Magical girl (Japanese: 魔法 少女, Hepburn: mahō shōjo) is a subgenre of Japanese fantasy light novels, manga, anime, and video games which features girls who use magic or possess magical powers. Magical girls transform to unlock their powers and are often accompanied by an animal mascot, using wands or scepters as a weapon to fight monsters and the forces of evil. 
History of the Genre
Manga and anime historians regard the Princess Knight manga, released in 1953, as the prototype for the magical girl genre. Himitsu no Akko-chan, serialized nine years later (1962) in Ribon, is generally accepted to be the earliest magical girl manga. Sally the Witch, adapted from the manga of the same name, is regarded by historians as the first magical girl anime. Sally the Witch was inspired by the Japanese dub of the television series Bewitched.
Originally, all Magical Girl shows were produced by Toei Animation, so "Magical Girl" wasn't so much a genre as a Series Franchise. This lasted until Ashi Production's Magical Princess Minky Momo hit the airwaves in 1982. The first instance of a magical girl team was a crossover in 1987 between Studio Pierrot's four '80s Magical Girl shows (Magical Angel Creamy Mami, Persia, the Magic Fairy, Magical Star Magical Emi, and Magical Idol Pastel Yumi. 
The Magical Girl Warrior subgenre, despite being the most well-known style of Magical Girl show in the west, didn't hit until Sailor Moon in 1992. This was essentially a combination of the earlier style shows with the superhero genre. Sailor Moon was a huge hit, and, naturally, other shows were made in the same style.
The wave of shows inspired by Sailor Moon eventually subsided, but new sub-genres spawned soon in its wake. As of present, most magical girl shows can be loosely organized into three broad categories.
Neo-classical, codified by Cardcaptor Sakura. Essentially, old school magical girl coming of age stories updated with the sensibilities of the modern age and the roles of girls and women in it. Mainly aimed towards young girls but often with a significant peripheral demographic of adult males. Contemporary examples include Ojamajo Doremi, Shugo Chara! and the Pretty Cure franchise (though that also fits in the second set).
Action Hero, created by Pretty Cure, but codified by Magical Girl Lyrical Nanoha. Magical Girl Warrior territory, with emphasis on "Warrior" and often enough Hot Blood to put a Shounen fighting series to shame. Largely aimed at the teenage and adult male demographic, and as such placing heavy emphasis on Fanservice. The Improbably Female Cast is frequently used as an excuse for Les Yay. Examples include: Mai Hi ME, Vividred Operation, and Senki Zesshou Symphogear.
Deconstructive, codified by Puella Magi Madoka Magica. Debunking the very concept of a world where young girls are forced to confront evil as a Crapsaccharine World with plenty of dark secrets and delving deep into the psychology of its cast, often with religious or philosophical references. Madoka itself was considered the equivalent of Neon Genesis Evangelion and Berserk for the genre. Other notable examples include Princess Tutu, Il Sole penetra le illusioni, and Yuki Yuna is a Hero.
Of course, there are other examples that feature similar themes but diverge even further from the old-style shows. Many fans felt that shows such as Magic Knight Rayearth were still Magical Girl shows, despite all the dissimilarities from the previous generation (others disagree, and feel that Rayearth is Shoujo RPG World Fantasy instead).