Just as the White Guardian is the embodiment of order, good, and chilling in a straw hat, the Black Guardian is the embodiment of chaos, evil, and cackling megalomaniacally, because racism.
Like all Guardians of Time (just the two of them on TV, but the novels add blue, red, pink, and orange ones named Inky, Blinky, Pinky, and Clyde), the Black Guardian is an immensely powerful being, a master of all reality, a god to the Eternals who are gods to the Time Lords, but he can't actually do shit except for whisper at Turlough or appear on the TARDIS scanner. He can wear mediocre disguises, though, which makes him almost as impressive as a human. He's beyond the need for technology, but he knows all about how the TARDIS worked and wants to steal it, and he tries to manipulate the Doctor into stealing the Key to Time, and so on.
The Black Guardian was so powerful and so dangerous that, after thwarting his plans, the Doctor had to fit a randomizer to the TARDIS to avoid being tracked by him and subjected to an unavoidable and horrific revenge... Until the Doctor got bored of that after a few weeks and got rid of the randomizer, and nothing happened.
Despite being named The Black Guardian, he was played by a white guy, Valentine Lyall, although occasionally in blueface. In the Big Finish audios, he's played by David Troughton; it's unknown whether David wears the blueface.