The blue and green counts are notifications of new activity in the Discourse forum.
Green notifications counts are considered high priority notifications and include private messages notifications and bookmark reminders notifications.
Everything else generates a blue notification, for example, mentions in a topic, new replies in a tracked topic, likes, badges granted, etc. Green notifications will persist when you click on your avatar menu and will only go away when you click on the notification or manually dismiss them, whereas blue are almost always dismissed as soon as you click the avatar menu.
If you are a moderator or an admin, you may receive a red notification. These are generated when there are items that require a review, like flagged posts, spam, etc.
You can see some notifications types in the code here.
And in this topic you can find a guide to Discourse’s interface nomenclature: Interface Nomenclature Guide