As you solve each case, you'll return to Lauren's apartment after unsuccessfully chasing the Countess. These little interludes are there for you to talk with Joey and try to figure out the Countess. All you have to do when you arrive back at the apartment is switch to Joey and talk to Lauren. Explore all of the dialogue options. Once you're done, you'll be able to switch back to Lauren and head out to the next case.
The Countess is a crazy old lady that will appear after you help each of the spirits to rest. Her first appearance will actually be just before you lay the first spirit to rest. Your conversations with her will be strange, to say the least. She is obviously completely unhinged. You should also keep your ears open when you lay each spirit to rest as the spirits will both mention a crazy old woman before heading into the light.
Saying any more about the interludes and the Countess at this point would potentially risk of ruining the story. This should be more than enough information for you to proceed smoothly through the game.