A high ISO lets you take pictures in low light, but produce more image noise, especially with small-sensor cameras like compacts and phone cameras. ISOs of 1600 or 3200 are mostly gimmicks and should be used only in an emergency where a bad picture is better than no picture at all.
This is why I suggested setting the ISO number manually so that the camera will not try to use a higher setting.
DSLRs have larger sensors and have less noise because the individual pixels on the sensor are larger. Moreover, the CMOS sensors they use are less noisy than the CCD sensors of compacts. Even then, as a general rule, it is better to go no higher than 800 unless lighting conditions make it necessary to use a higher ISO.
With point-and-shoot cameras, noise is usually noticeable at ISO 200 and gets worse at higher settings. The effect is less in good light but it's still there.
The disadvantage of using low ISO is that it needs a longer exposure (slow shutter speed). In low light, this can produce a blurred image due to camera shake or subject movement.
Going back to moon shots, most inexperienced photographers think they need a high exposure because the rest of the scene is dark. But the surface of the moon is as bright as a sunlit landscape here on earth. So exposure requirements are approximately the same as a daylight shot, though pollution and haze can change that.
The camera thinks the same way as an inexperienced photographer if it is set for auto-exposure with scene-average metering. That's why we have to use manual settings or spot metering.
Spot metering measures the amount of light in a small selected area. With simple P&S cameras, the selected area is in the centre, while advanced cameras offer the option of choosing other spots in the scene.