Pretty simple. On/off CR is the greatest difference in lumen levels from a fully white image and a fully black image. (literally as bright as the projector can go to as dark as it can go). *** edited to include the caveat: when properly calibrated as it should be for optimal viewing. Of course, you could also measure the 'out of the box' values.***
ANSI CR is a measurement taken from a specific screen which is divided into 16 squares. The 4 x 4 grid has alternating black and white squares and the CR is measured as the difference between the average lumens measured from the centers of the white squares to the average lumens measured from the centers of all the black squares.
On/off CR gives you the greatest possible difference between the brightest and darkest a projector can project between different frames whilst ANSI CR gves you the greatest difference within a single frame/image. All projector manufacturers list on/off CR. And while projectors can achieve a 5000:1 on/off CR or higher, ANSI CR is not only limited by the projector and technology but the surrounding environment too. With white walls and a white ceiling, it is pretty much impossible to reach even 100:1 ANSI CR because the reflected light from the brightest parts of the image will wash out the darkest. To enjoy maximum ANSI CR, you would have to blackout the room to not only cut out ambient light, but to absorb ambient light reflections off the screen from being reflected back to the screen.