Stands for "Natural User Interface." An NUI is a type of user interface that is designed to feel as natural as possible to the user. The goal of an NUI is to create seamless interaction between the human and machine, making the interface itself seem to disappear.
A common example of a natural user interface is a touchscreen interface, which allows you to move and manipulate objects by tapping and dragging your finger(s) on the screen. The digital objects on the screen respond to your touch, much like physical objects would. This direct feedback provided by a touchscreen interface makes it seem more natural than using a keyboard and mouse to interact with the objects on the screen.
Another modern example of an NUI is a motion-based video game. The Nintendo Wii, for instance, allows you to wave a controller in the air to perform actions on the screen. Microsoft's Xbox Kinect allows you to control your on-screen character by simply moving your body. Both of these motion-based interfaces are considered natural user interfaces since they respond to your natural motions.
While touchscreens and motion-based games are two of the most common types of NUIs, several others exist as well. For example, a voice recognition interface like Apple's Siri assistant on the iPhone is considered a natural user interface since it responds to naturally spoken commands and questions. Virtual reality devices are NUIs, since they emulate a real world experience. Even some robots are considered natural user interfaces since they respond to human motion and spoken commands.