In order to show the world that minority characters are not bad people, one will step forward to help a "normal" person, with their pure heart and Closer To Earth wisdom. They are from a minority that is discriminated against, physically or mentally disabled, or social outcasts (drifters, the homeless, ex-cons). They step (often clad in a clean, white suit) into the life of the central character (often white, American and racist) and, in some way, enrich the central character's life.
While this can work as a plea for tolerance, or simply An Aesop about not dismissing people just because they're different, it's all too easy to go too far and make them into an all-knowing Mary Sue or pseudo-narrator whose magical minority-powers save the day. It also tends to raise the question that if the Magical Negro is so powerful and intelligent, why he's never saving the day, himself, instead of helping the mainstream hero to get all the glory. Also, quite often he's just ditched off or even killed after he's fulfilled his purpose for the plot.
If the Magical Negro is from a society of Noble Savages, expect an Anvilicious Aesop about the failings of the society which protagonist comes from - which usually leads to the protagonist 'going native' and ending up better at everything than his Magical Negro mentor.
The reason the Magical Negro is problematic is because it is a moral and artistic shortcut, replacing a genuine moral message with a well-intentioned but patronizing homage to the special gifts of the meek.
Morgan Freeman is the ultimate Magical Negro actor. Referenced on The Daily Show when discussing Barack Obama's need to pick a running mate who won't just turn him into one.
See also Whoopi Epiphany Speech and Black Best Friend. For a similar trope about women, see Manic Pixie Dream Girl.