To chat, hit ':', type your message, and hit Return. By default, that will send a message to the #public channel.
You can send private messages to other players or to parties. Begin your message with the target name and a colon,
Bilbo: what's in your pocket?
You can use short names, up to the point when there is no ambiguity. For example,
bil: well, what is it?
will resolve to Bilbo, unless there's another online player or a party starting with bil. Even one letter should suffice.
The game will never send your private message to the wrong recipient in case of ambiguity, it will return an error instead.
When you enter the game, your default chat channel is #public. To change channels, type #channel_name into chat prompt, without any messages.
For example,
#hobbits
To send message to another channel without changing it, prepend channel name to your message, like so:
#public Cheers, lads
There are several "virtual" channels, that you don't need to join, yet can send messages to.
They are &say and &yell. The first one can be used to talk to people in immediate vicinity (in your line of sight). The '&yell' channel is heard by all people on the same dungeon level as you. In addition, it will disturb players in your immediate vicinity.
You address those the same way you would address private messages, by prepending channel name and a colon to the beginning of the message. For example:
&say: Hello
or
&yell: WAIT!!!
If you have a window dedicated to chat messages, you will see it has a "Chat Tab" bar on the top. By default, you shall only see '#public' channel. But you can easily add more chat tabs and switch between them.
To control the chat tab interface, press the ' key. It will then present you several sub-commands, for switching between tabs (p, n), closing tabs (c), and creating new ones (o).
When creating new tabs, you can type anything you would normally use in your chat prompt, for example chat tab #hobbits will display the chat channel with that name, and the Bilbo tab will display private messages from/to that player.
Note: don't forget that you can macro those functions. If you have some keys to spare, \e'p and \e'n can make convinient shortcuts for 'previous tab' and 'next tab'.