Here's a good example of how the config values (N,U,T) work. Pick up a pen, pencil, etc and point it straight out in front of you with your elbow high in the air. Then, without moving the tip of the pen or rotating its orientation, drop your elbow down.
The pen's location in space and orientation hasn't changed, but your elbow and wrist joint "values" have. In the case of 6 axis robotics, there's 3 main scenarios that can cause this:
Non-Flip or Flip: Joints 4, 5, and 6 rotate to "flip" the wrist over, maintaining the Tool Center Point in the same location and orientation. This would be like rotating your wrist and somehow being able to rotate your fingers separately in the opposite direction to maintain the pen in the same location and orientation; this one is physically impossible for a human to replicate naturally.
Up or Down: Joints 2, 3, and 5 rotate to raise or lower the "elbow" of the robot (similar to the above example I mentioned, maintaining the Tool Center Point in the same location and orientation.
fronT or Back: The robot rotates J1 roughly 180 degrees, flips Joints 2 and 3 above over the back of the robot, and rotates Joints 4, 5, and 6 to maintain the same Tool Center Point location and orientation. This is like turning around with your back to your target, reaching behind your back over your shoulder, and rotating your wrist to bring the pen to the same location and orientation.