The Info Button can be pressed to display the Information Display Screen on the rear LCD Monitor, where you can view your shooting and camera settings, etc. If you then press the i Button, you will "activate" the Info screen, and then you can change the settings along the bottom of the screen, such as noise reduction, ADL, bracketing, and some of the button assignments.
You can also just press the i Button to turn on the Information screen and have it immediately "activated."
When in Live View, you can press the i Button to access several applicable, contextual options, such as ADL, Electronic Front Curtain Shutter, Split-screen Zoom, and Live View WB, where you can set the WB of the monitor to be different that the camera's WB setting. In LV Movie, the options include movie quality, microphone settings, and zebra stripes.
Several of these i Button options are somewhat "hidden" features because it is the only way to access them. I wrote a bit about them in a blog post about the D810 "hidden" features:
http://blog.dojoklo.com/2014/07/17/hidden-features-of-the-nikon-d810/
Pressing the Info button in LV or Movie will scroll through the various display screens, such as grid, electronic level, shooting information on or off.
If you press the i Button during playback, you can access the Retouch menu. Press it during movie playback to access Movie Editing.