A network "port" may sound like a physical connection of some kind, like the USB ports on the back (or front or side) of your computer. In network terminology, though, a port is most often a logical or virtual concept, part of a network transport protocol (TCP/IP, for example, or UDP). In this sense, a port is a logical destination on a specific host that is identified with some particular service or listener. Port 21, for example, is often used for FTP servers.
Ports are most often significant to FileMaker administrators who administer or otherwise encounter firewalls. Many firewalls do port-based blocking or filtering of traffic. If traffic to or from a particular port is not permitted across a firewall, services may be disrupted or blocked. For example, if port 21 is blocked on a firewall, an FTP server behind that firewall will be unreachable. If port 5003 is blocked on a firewall, a FileMaker server behind the firewall will be unreachable.
For an extensive listing of port numbers, see the list maintained by IANA at http://www.iana.org/assignments/port-numbers. Some of the FileMaker network ports are registered whereas others are not, though this has little practical significance for FileMaker administrators.