What is the black box called by battery under hood of a 2001 Mercury Cougar?

That's likely the Powertrain Control Module (PCM), sometimes also called the Engine Control Module (ECM). While it's not strictly a "black box," it's a dark-colored, sealed electronic control unit that's crucial for the car's engine and transmission operation. It's often located near the battery, though the exact location can vary slightly depending on the specific model year and trim.