The global inbox was once introduced because it became hard "to keep track of all your questions, answers, and comments across every site you participated on".
The small red number tells you how many new replies you have across the entire Stack Exchange network of websites. And by replies, I mean:
- New answers to your questions
- New comments on your posts
- @replies to you in comments
However, it's been a while since 2010, and there are many more events triggering an inbox notification. What kind of notifications are there nowadays?