You will be notified by a sound and flashing message on the browser tab when you receive a new chat, and if you use Gmail,  a chat window will open automatically, allowing you to read the chat message and reply as required.  You can change notification settings as required.

In both Gmail and the Chat app, the person that has sent you a message will show an indicator, and a sound will play. 

General chat settings

To change notification settings:

Choose to allow notifications, then the sound to play when a notification is received. You can also choose to  recive an email notification for @mentions and direct messages.

Click on the X in the top-left corner of the screen to close the Settings window when complete.

Conversation and space notifications

For each conversation in Chat, you can choose to turn notifications off by positioning the mouse over the conversation, clicking on the ellipses, then choosing Turn notifcaitons off.

You can change notification settings for each space that you are a member of, to choose how you are notified of each new message.