The cranes take one container off at a time (except when the container underneath sticks to the top one, which is not good). Actually, wikipedia says some cranes do two to four containers on purpose now.

Usually the crane operator will load/unload several containers without moving along the dock; because they're all lined up, the crane only needs to travel in two dimensions (vertically and across the boat from port to starboard). The cranes are specialized to pick up containers by the top corners, which makes connecting fast. Standardized containers means the corners are at the same place (ok, there's a few sizes, but 40 ft containers are the vast majority of ocean shipping) and lining up is easy. On the dockside, there's a crew of longshoremen that move chassis (trailers) into place for the crane to drop (gently, usually) the containers on, those are then parked nearby, etc.

Depending on the ship (and the dockside staffing), you can have multiple cranes working the ship. Planning is required to keep the ship balanced and minimize the number of containers moved. These ships generally visit several ports in order, and usually both unload and load at all of them, so it's complex.

Depending on the traffic (and pandemics), the port runs up to three shifts.