It's difficult to definitively say what the "second major structure" of the brain is because there are many critical components. It depends on what criteria you're using to define "major."

Here are a few possibilities, along with their significance:

* Cerebellum: This is often considered the second largest part of the brain after the cerebrum. It plays a crucial role in motor control, coordination, balance, and learning motor skills.

* Brainstem: This structure connects the cerebrum and cerebellum to the spinal cord. It controls essential functions like breathing, heart rate, and blood pressure.

* Limbic System: This is a collection of interconnected structures deep within the brain that is crucial for emotions, memory, and motivation. It includes the amygdala, hippocampus, and hypothalamus.

* Thalamus: This structure acts as a relay center for sensory information, filtering and sending it to the appropriate areas of the cerebrum.

Ultimately, the "second most important" structure depends on what you're focusing on. Each of these structures is essential for different aspects of brain function.