Push notifications will popup in your browser when you are logged into H-M. So you can be watching a youtube video, or reading a forum and the browser will notify you with a popup. (EDIT: the popup is in the lower corner of your monitor.)
This is a preference that you set. Go to your profile and turn it off or on depending on whether you find them useful or not.
Yes, some people see them as no different than ads...others find them very useful if they are tracking a forum or question that they have asked and want to jump right on it when there is a reply, for example.
Anyway, your preference. It is not a requirement of the site to receive push notifications.