On twitter's main desktop web site, there are 4 sort-of buttons at the top: Home, Connect, Discover and Me. Home sort of illuminates blue when there are unread/unloaded tweets, and Connect illuminates when you have new followers or new mentions. Does anyone know why the "Me" button illuminates?
Aha! Apparently there's a button on your "me" tab for direct messages, letting you know you have ones you haven't viewed. Seems to be separate from the direct messages link on the gear icon at the top of the page.