How to turn off flash in my Nikon D5100?

There seems to be no Flash Off option in the camera. There's a No Flash mode on the dial, but that's like Auto mode without flash. You can't use any other features with it.

Just don't pop the flash up and it's off. If you want to use an "auto" mode that doesn't force the flash up, use "P" mode.

Should also be a menu option for it. Read your manual.