Ghosts haunt an underwater facility, the crew fight for their survival, and the Doctor tries to solve the mystery of their dead friend/family member/pet.
In 22nd Century Scotland, a mining crew digs for oil under a lake that is in fact a flooded town. After discovering an ancient spaceship on the lakebed, they bring it on board their to investigate its origin. Little did they know they also brought the ghost of an alien that wastes no time in trying to kill the crew.
This is the situation that an unhappy TARDIS has brought the Doctor and Clara. Shortly after their arrival a few days after the initial haunting, the ghosts greet them in the facility by showing them the ship and the strange letters carved into it, and then try to kill them. They manage to escape the ghosts and hide in the base's Faraday cage, used in case of a leak from the base's nuclear reactor.
Together, The Doctor, Clara, and the mining crew, lead by Cass, a deaf woman, and her signer Lunn, must figure out what the ghosts want, and why they are there in the first place.
"Under the Lake" shares the tone of many episodes seen during Russell T. Davies days as showrunner. The Doctor and companion come to a location they were not prepared for, and are swept up in a mystery. Specifically, Nine's adventures in "The Unquiet Dead" and "The Empty Child" come to mind.
This style of episode is less universe or planet saving with dire galactic consequences against an ancient foe, and more a smaller story revolving around a group of characters, and while the implications of the crisis could spread to the rest of the world, it's really these few characters lives that are at stake. It's a nice change of pace from Steven Moffat's usual grander episodes.
Also, This is the second time in three episodes that Clara has mentioned some racy, possibly self destructive behavior in her introduction to the episode. In "The Magician's Apprentice" she mentions Jane Austen being a fantastic kisser. Here, she mentions leaving her sunglasses and most of her dignity on a planet that celebrates New Years for two years. Even the Doctor notices something is not quite right with his companion, worrying she will go native. "There's an entire dimension in this room, but only one me," says the Doctor. He seems worried about her, and that, like him, she is running from her problems.
(I could be looking at this too closely, but that's the sense I got.)