Penumbra: Black Plague picks up the story right where Penumbra: Overture left off. And given that Overture was a pretty short game that ended in a cliffhanger, I think it's fair to say that a sequel was required. It's a quick ride to what is basically the end of the Penumbra story (although Penumbra: Requiem does flesh things out a bit more), clocking in at somewhere between 5 and 10 hours of gameplay. If you're in a real hurry and follow a walkthrough, you could probably finish in even less time than that, though.
In Black Plague, you wake up after being knocked unconscious at the end of Overture and continue working your way through the facility to try to better understand your father's secrets and the mystery at the heart of the series. The story is very Lovecraftian; as a big fan of some of his stories, I felt particularly drawn to it. And like many Lovecraft stories, this won't let you down. The story is what really drives this game. Much like in BioShock, Doom 3, and Dead Space, the history of the facility is largely revealed through scraps of documents or computer entries found scattered throughout the levels. Your interaction with other characters is pretty minimal, mostly consisting of a very caustic split personality that develops early on in the game and a woman you meet through radio and video equipment scattered throughout the environment. Like Red in Overture, much of the game is spent trying to reach her, and when you finally do the ending is not happy.
At heart, Black Plague i a puzzle game, moreso than Overture was. The parts of Overture I found least satisfying (sneaking around a lot, and of course the fairly awful combat system) have been minimized here. Black Plague still provides plenty of ways to get yourself killed, and manages to maintain a sense of horror throughout, but does so without having to resort to cheap deaths via a clumsy combat interface. The puzzles, like in Overture, remain challenging but without resorting to some of the ridiculous solutions that more traditional adventure games are known for (Old Man Murray had a great article relating to this some year's ago, located here). All in all, the whole series is worth playing through, but Black Plague is probably the best of the three.