It is possible to get notified when someone in your organization performs an action. You can subscribe to cards, columns, backlogs and boards.
You can easily mark all notifications as read by clicking on Mark all as read when viewing notifications. Clicking on a notification will mark it as read and open the card.
By default, you will be automatically notified about cards you are assigned to. You can disable this on the Notifications tab under Edit Profile.
If you haven't logged in for a while, you will also be notified via email.
You can read more about notifications and newsfeed here.