The ContentManagerName property returns the Screen's ContentManagerName. For more information on ContentManagers, see the ContentManager page.
ContentManagerName can be used when manually loading files. For example, assuming that "Content/MyGraphic.png" is part of your project, you can load it as follows:
// Assuming "this" is a screen (the code is in the Screen's custom code:
Texture2D texture = FlatRedBallServices.Load<Texture2D>(
"Content/MyGraphic.png", this.ContentManagerName);