The surface of the Moon is about as reflective as an asphalt road--rather dark but not totally black. When we look at the Moon, we are seeing the light which reflects off it. The moon is just as likely to be visible during the day as it is at night — it orbits Earth independently of the sun. When its orbit brings it to our part of the sky during daylight hours, it is illuminated by the sun, and we can see it.
It might be useful to think of the Sun as a large light bulb, and the moon as a large mirror. There are situations where we can't see the light bulb, but we can see the light from the bulb reflected in the mirror. This is the situation when the moon is out at night. We can't see the Sun directly because the earth is blocking our view of it, but we can see its light reflected from the moon. However, there are also situations as mentioned above where we can see both the light bulb and the mirror, and this is what is happening when we see the moon during the day.