Manual mode: the flash fires at its full power. You control the amount of exposure on your film by setting the f-stop on your camera. The chart helps you to determine the f-stop to use. The shutter speed is irrelevant to the flash exposure that the film will sense, but ambient (non-flash light) will still expose the film. So, for example, if you are outdoors, using ASA 800 film and a flash sync speed of 1/60 in order to fill-flash a person standing 20 feet from the flash, you will probably over expose the film even if you do set the flash correctly.
Auto mode: the flash has a crude, non-directional sensor on the front of it. This sensor must be unobstructed and pointing at the subject. It is handy, but not always accurate if you are finicky. This flash obviously has two flash intensities that it will use in auto mode. The higher intensity (green) will result in better flash-subject distances, smaller f-stop settings on the camera, and shorter flash durations which might be better for stop-action. It will also probably use up your batteries faster and age the capacitor more. The lower intensity will allow you to use the flash at closer distances or will give you a bit more flexibility in adjusting your fill-flash. Auto modes rely on you or your camera telling the flash what film speed you are using and what the aperture is set to. If you look around you will probably find a way to do that, a dial or a slide switch or something. Or it may rely on an extra pin to get that information from the camera, in which case it would be specific to a particular brand of camera. I have no idea if it is TTL flash capable.
Distance: the distances in this chart are flash-to-subject distances, not camera to subject. It is irrelevant how far your camera is from the subject. You are capturing the light that is reflected from your subject, so what matters is how much light is falling on your subject.
Chart: the full chart is used for manual mode, including the green and red columns. It's just a simple falloff chart. You can even extrapolate the chart out to the right or down to faster film speeds.