The way HeadsUp works is very simple: once you have it activated, instead of seeing notifications in the status bar as usual, you'll see several small white panels. These display more complete notifications, and even let you do some actions directly (like sharing if it's a photo, responding if it's a message, etc).
In the HeadsUp options, you can create a blacklist of apps that won't appear in your notifications. This is especially useful for certain instant messaging applications, or even for video games.
HeadsUp provides a very interesting alternative to traditional ways of receiving notifications. It is not only more aesthetically pleasing, but also more practical.