The alliance chat is used to communicate with other players that are in your alliance. As such, only the members of your alliance that are online will be able to read what is said here.
Talking[]
The chat box has an option to choose whether to speak in your alliance chat. Click on the rectangular button in the bottom left of the chat box and choose "Alliance". The button will now change to show the abbreviation "All." Your chat will now start with [Alliance chat] and will now be a turquoise color. It costs no horns to talk in the alliance chat.
There is an "Alliance" button at the top of the chat box as well. Select this to show all of the alliance chat you have received while online.