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Jabber is now officially called 'XMPP'. I prefer to call it Jabber as the name is much friendly than XMPP.
Jabber is a decentralized instant messaging (IM) service, meaning that there is no central server regulating it's use.
This is different from popular proprietary IM protocols such as AIM, ICQ, MSN Messenger, and Yahoo Messenger, which all must communicate (or at least establish a connection) via a central server.
Jabber uses a feature called server-to-server communications. A Jabber user who wants to send a message to another user contacts his own XMPP server, which relays the message to the recipient's server.
For this reason, Jabber addresses mimic email addresses, with each part of the address serving the same purpose as it's counterpart in an email address. Jabber IM accounts can “look” different - there is no “@jabber.” syntax. Think of the differences with e-mail addresses - @hotmail.com, @gmail.com, @ei8fdb.org. Jabber accounts can communicate the same way these e-mail addresses can , server to server.
There are many Jabber servers available which provide free access to people who want to create a Jabber account.
This is a difficult question. Personally I use jabber.ccc.de. The only recommendation I would make is to not use GTalk, as Google have removed the server-to-server feature from GTalk.
This means GTalk users are not able to communicate with people using other Jabber servers. This is the same as Google not allowing people with GMail accounts to communicate via e-mail with friends using Hotmail, or Yahoo, or ei8fdb.org.
Extensible Messaging and Presence Protocol (XMPP) is a communications protocol for message-oriented middleware based on XML (Extensible Markup Language). The protocol was originally named Jabber, and was developed by the Jabber open-source community in 1999 for near real-time, instant messaging (IM), presence information, and contact list maintenance.
Designed to be extensible, the protocol has also been used for publish-subscribe systems, signalling for VoIP, video, file transfer, gaming, Internet of Things applications such as the smart grid, and social networking services.
(This is an extract from the Wikipedia article on XMPP.)
Good question! Here is a list of software applications that support Jabber accounts.