When a new chat comes in it will be shown to all available agents and be accompanied by an audio signal. As an agent you have two possibilities. You can either Accept or Decline. As soon as an available agent accepts the new chat, it will be removed from all remaining agent screens.
Once accepted you can respond to the request and continue the conversation. If you Decline the chat request, it will be removed from your screen but will still be available to all other available agents.