This lovely disc has a few names - wall protector, wall bumper, door bumper, door knob protector, and so on. The point being that you protect your wall (behind a door) from the knob going through your sheetrock.
These little doo-dads are self-adhesive and attach easily to the wall. I like that they are flush to the wall and paintable. I have a couple in my own house.
If you've just had painting or wallpapering done and you want to protect a wall behind a door, I would suggest spending the $3.00 to prevent damage that could result in a major repair. But please, paint them or put a piece of the wallpaper that is in the room where you are placing them, on top.
It's paintable so that you can integrate it into your wall decor. You've put time and money into painting or wallpapering, so protect it.