That box is part of the air intake system. It's where the air filter should be and it should have a tube running from it to other box (top center of your photo - it's a hat for the throttle body). You can see how it should look in this photo:
I'm assuming you took the lid off of your air box, disconnected it from the throttle body hat and what you really want to know is what the tube on the inside connects to. The large tube does not attach to anything inside the lid. It just provides a path for the air to transition into the intake tube after it goes through the air filter.
There should also be another tube connected to the throttle body hat (smaller tube to the left of the larger one). This, I believe, should be going to PCV (Positive Crankcase Ventilation) system. I don't see it in your picture, so wonder where it could be at. It is important in how the engine runs and behaves, so you probably want to ensure it gets put back. You can see it in the picture I've provided.