The term Port Calls describes events that take place in ports such as arrivals and departures of vessels. The Port Calls Box displays the most recent such events, seconds after they have happened. It keeps automatically updating so that it contains the most recent events at any given moment.
Each row describes a Port Call - you can see the name of the port and its location (the small flag next to its name), the Port Call type (arrival-departure), the elapsed time from the moment of the Port Call and the name (and type) of the vessel that took part in it (as well as the flag she is under).
The Port Calls Box is located on the MarineTraffic homepage just below the Live Map area and next to the Latest Photos Box.
In case you have created your own fleet(s), you are able to view the most recent Port Calls that are recorded by the vessels that are part of it (or them, if you have created multiple fleets). All you have to do is to simply alternate between the My Fleets and Any tab from the option buttons that are located on the top right corner of the Port Calls Box.