The ‘Inbox’ is just where the website puts any incoming email notifications/messages.
How fast ones forum account Inbox fills up will of course vary from user to user but I think we can keep up to 100 notifications in there, after that you will need to start deleting some. I would generally start with the oldest ones first when it becomes necessary.

As you will no doubt remember back in the day, in the real world of offices and desks, people often had two physical trays, one for correspondences yet to be looked at and one for things that were to go out. And though more usually applied to visual content, this digital mimicking of real world characteristics actually has a name; Skeuomorphism!