'External Connections (EC) is a bi-directional interface aMule uses to communicate with external utilities, specifically:
When a user is using one of these programs, it is sending the commands via the External Connections port and aMule is reading them there.
Unless aMule and the external utilities are used on the same machine, you may need to configure your firewall and/or router to manage the ports properly, ie, open them to incoming traffic.
The Firewall article has instructions on how to do this.
As of December 2006, aMule SVN has experimental code enabling Universal Plug and Play (uPnP), which enables you to configure the ports on a uPnP-compatible router automatically. This functionality should soon be available in a general release.
NOTE for SVN users: It is very important that the aMuleWeb, aMuleCMD, and aMuleGUI you use are built from the same daily CVS tarball as aMule; mixing versions you are liable to experience crashes. For official releases, the developers take extra care that the worst that happens when you mix versions is a message "Invalid protocol version".
NOTE: There are alternate ways of communicating with aMule although External Connections is the only bi-directional way. Other ways of communication with aMule would be the Online Signature (outgoing direction) and ED2KLinks file (ingoing direction).
For more information on the EC protocol, see External Connections v2.