The cerebellum, or "little brain" as its name suggests, sits at the back of the brainstem underneath the cerebrum. As you can see, it really does look like a smaller version of the cerebrum, hence its name. Its main job is integrating motor functions. It does not initiate any motor movement on its own, but rather coordinates those signals and integrates the output. For example, when you stand up straight, you must coordinate many different muscles across your body to maintain your posture. Without the cerebellum, you would simply face plant, unable to maintain balance. It also plays a role in fine motor movements, such as performing delicate tasks with your fingers.
Moving on, the brain stem contains the pons and the medulla oblongata, which are located right next to the cerebellum. However, our image here lacks a cerebellum. I don't know why. Perhaps there is a bias, maybe emphasizing the brain stem. The pons and the medulla oblongata have roles in unconscious types of functions that are necessary for life. These structures are often considered as parts of the "lizard brain," responsible for very basal functions that are absolutely essential for sustaining life. These areas are crucial; for instance, the pons is involved in swallowing, breathing, eye movements, posture, and notably, in dream production, which is not well understood at all. Sleep and dreaming remain significant mysteries in science, but it is known that the pons plays an important role in producing dreams.
The medulla oblongata also manages many passive unconscious functions necessary for life, including maintaining heart rate. It also plays roles in breathing, alongside the pons, and in regulating blood pressure. Therefore, these structures are often thought of as more basal because they handle those unconscious functions. So with that, let's actually go ahead and flip the page.