There are two types of notifications in Teamwork Chat. The first is when you are mentioned directly. You will see that the notifications have numbers to indicate how many unread mentions you have.

The second notification is a red dot. This means that there are messages in chat that have not been read. Messages that you have not been mentioned in. If you look to all of the conversations in the left hand pane of chat, the room will be in bold white to show there is an unread message.