Natural User Interfaces (NUI) still lacks a formal definition. It can appear as a synonym for intuitive, easy or gesture based interaction. So rather than trying to define one I will list some characteristics that are usually associated with them:
If you are new to the term, this is a cool video that has some examples and attempts to define some concepts:
In video games, the use of more natural interfaces helped technology intimidated people to get the courage to jump in and play. Even if most hard-core gamers still prefer classic game controllers, these new interaction techniques really helped to expand the market for videogame consoles. Bowling with the Wiimote seems more natural and simple than using a combination of buttons and analog sticks. The Nintendo Wii, Playstation Move and Microsoft Kinect are successful technologies that opened the path to development of more natural interactions for videogames.
It is hard to talk about an absolute natural interface, but the previous example shows that one interface can feel more natural than another. The extent of this feeling depends not only on the technology but also on the activity being performed and the actual user of the interface. This further complicate the issue since NUI relies on previous background knowledge about how things are supposed to work. This assumptions are ultimately related to culture and previous experience of each user.
With a few exceptions most user interfaces are a coolest but not so efficient way to perform a task. If we look closely we can see that in few examples, some actions performed are far from being natural. Even int the coolest complex hand/head/spirit tracking setup. In fact, a natural interaction does not require complex multiple degrees of freedom setup. Its quality emerge from the perfect coupling between the hardware, the software, the task and the user.
More on that latter. Or not.