I'm not sure what its called but I'm looking to write a Lightning Component which utilizes the below image's section of Lightning App Builder. Does anyone know what its called? Doesn't have to be rich text, a simple text box in that section would do just fine.
This is outlined in Configure Components for Lightning Pages and the Lightning App Builder. Basically, you add a "design" resource, which allows developers to expose text boxes, Boolean values, and picklists to administrators for customization. You can read more about that in Aura Component Bundle Design Resources.