What are the five most important structures in the brain, and why?
I am assuming that you are talking about the human brain. Most mammalian brains are similar in terms of organization, they mostly differ in arrangement and size. While there are no universally agreed upon five most important structures in the brain, I will highlight what I believe to be the five most significant structures in the human brain.
The largest single structure in the brain that takes up most of its volume is the cerebrum. It is divided into four major hemispheres, or lobes, named the frontal, parietal, temporal, and occipital lobes. The cerebrum is responsible for a huge variety of human activity including but not limited to sight, memory and learning, speech, hearing, behavior, and spacial perception.
The cerebrum is surrounded by a thin layer of nerve tissue with lots of surface area called the cerebral cortex. This area is gray colored, hence the term "gray matter" when referring to the brain, although most of the brain is not gray colored. The cerebral cortex is involved with processing sensory information, voluntary movements, and what is called association (which means perception and interaction with the environment around you).
The bottom rear of the brain contains the brainstem, which itself is composed of smaller subunits but we will consider the brainstem as a whole. This connects the brain to the top of the spinal column and is responsible for much of the involuntary nervous system (things like breathing, heartbeat, sleeping, etc.)
The cerebellum sits just beneath the cerebrum. It plays an important part of basic motor control, or the movement of different parts of the body. It does not control movement directly, but it is responsible for coordinating multiple movements. As such it is important for physical equilibrium (balance), coordination, and the general timing of the body's movements.
Finally, the hypothalamus is a very small but important structure near the center of the brain. It is responsible for controlling the endocrine system via the pituitary gland. In this way, the hypothalamus controls hormone production and levels that affect a wide variety of bodily functions like growth, digestion, and bodily cycles (like sleep).