The alliance chat is a private chat channel for you and your alliance's members. You will need to join an alliance before you can access this chat window.
Once active, you can swipe left and right between global and alliance chat with your main chat bar.
You will need to have your alliance chat enabled via the in-game settings menu in order to see your alliance chat. If chat is enabled on your account, the blue square that you can tap to enable the feature should appear to be filled in completely.
