You'll get a notification in your Inbox when there are updates to tasks you care about / are subscribed to. Specifically, if you get assigned to a task, @mentioned on a task, or someone replies to your messages, you will be notified when there are updates.
Notifications to your Inbox are real-time — so as soon as there's an update, your Inbox will be badged.
Notifications will stay in your Inbox until you archive them (CMD+E ), but you can always see archived conversations from the ... menu if you want to find one you've already archived.
You can also unsubscribe from tasks if you no longer want to be notified when there are new updates or customize your notifications to choose which updates you want to be notified about.