According to Accuweather's page on the Anatomy of a Hurricane,
3. The Eyewall. A band of clouds, strong winds and heavy rains surrounding the eye of the storm. At the eyewall, there is rapid movement of air toward the center and upward into the cloud.
4. The Eye. What goes up must come down, so with the violent rising air converging toward the storm center at the eye, sinking air develops within. This air dries out, creating the clear, calm eye. Winds are very light here since the focus of convergence and hence strong winds are in the eyewall.
They also explain that the vacuum effect created by the rising air in the eye wall works like a toilet plunger on the ocean, lifting the water surface, and this creates the notorious storm surge, which is where the worst destruction of a hurricane is done.
The images of hurricane eyewalls were taken from the following linked pages...