What is a Puella Magi?

Puella Magi Madoka Magica is a show created by SHAFT. It's about a girl named Homura, who goes back in time to save Madoka, a Magical Girl who's fate is to die an inescapable death. With friends Sayaka, Mami, and Kyoko helping out, Homura was able to prevent Madoka's horrible fate, and gave her a beautiful one.

There are also PMMM spinoffs, such as Puella Magi Kazumi Magica and Puella Magi Oriko Magica. These feature some new characters, such as Kazumi and Yuma.

A Puella Magi is a girl who has her wish granted by an alien known as an Incubator. She gets her wish granted, powers, and the ability to see and fight witches with different types of magic.
There are different types of magic, given to you based on your wish. Time Magic, Enchantment Magic, and Weapon Magic to name a few.