No special privileges are required. Any remote host can connect to a server that's listening on a port below 1024. Furthermore, multiple simultaneous connections may be made to a remote host on a remote port. In short, no more than one process on the local host can use a port at one time. However many remot hosts may connect to the same remote port.
Many services run on well-known ports.
This means that the protocol specifies that the service should or must use a particular port. For example http servers generally listen on port 80. SMTP servers listen for connections on port 25. Echo servers listen on port 7. Discard servers listen on port 9.