The notifications icon (red bullet) indicates a new notification you have in Activity. By default, users receive push notifications when someone recommends a recipe to you, comments or performs an action on one of your recipes, or @mentions you. You'll also receive notifications regarding new seasonal content, feature tutorials, and updates to BigOven.
To remove this icon, simply launch the app and toggle to and from the Activity screen. To disable BigOven notifications altogether, visit your general device settings, tap into to the BigOven section, and switch the Notifications option to "Off."