Notifications
When someone follows your account or requests to follow you, mentions your user name, or boosts or favourites one of your toots, you will receive a notification for this. These will appear as desktop notifications on your computer (if your web browser supports this and you've enabled them) as well as in your "Notifications" column.
You can filter what kind of notifications you see in the Notifications column by clicking the Notification Settings icon at the top of the column and ticking or un-ticking what you do or don't want to see notifications for.
If your notifications become cluttered, you can clear the column by clicking the Clear icon at the top of the column; this will wipe its contents.
You can also disable notifications from people you don't follow or who don't follow you entirely - to do this, click the Preferences icon in the Compose column, select "Preferences" on the left-hand menu and check either of the respective "Block notifications" options.