You normally have only one physical connection to a network at a time, but often use it for more than one thing, or logical connection, at a time. Ports are used to tell the operating system, which logical connection the received data packets are intended for.
For example, web browsing with Hyper Text Transfer Protocol (HTTP) usually uses port 80 while File Transfer Protocol (FTP) normally uses port 21. When you request a webpage, your computer sends a request packet to port 80 on the server, as it expects the web site's server to wait for your call at the default port. Port numbers run from 0 to 65535. Some of them are "reserved" for official purposes. You can go to http://ports.tantalo.net/ , the official IANA port list and http://en.wikipedia.org/wiki/List_of_well_known_ports_(computing) to find out more about which ports normally belong to which protocol/program.
The general advice is to use a randomly selected port between 49160-65534. Please read this article for more information on which ports to use with Azureus.
Originally bittorrent implementations (like Vuze/Azureus) used the port 6881, but as some ISPs started to block that port, Vuze currently recommends to use some other port and it randomly selects a port for a new installation. The selected port number itself does not change the functionality, as long as you configure your own network accordingly. Read all about port forwarding.
Information about ports that Vuze reserves by default.
Read the Azureus FAQ