The moon is visible in daylight nearly every day, the exceptions being close to the new moon, when the moon is too close to the sun to be visible, and close to the full moon, when it is only visible at night.
The best time to observe the moon is just before dawn or just after dusk. At these times, its rays are still slanting across the sky from west to east, so it will appear to be rising or setting, respectively.
The moon is also visible during midday, but then it is not recognizable as such because it is so bright out. The best place to look for it is in the south, where it will be surrounded by a ring of bright stars - this is called "the glow" and it's what allows you to see it during the day.
The moon is also visible during twilight periods, but it is usually difficult to recognize because it is so bright compared to the dimming effect of the clouds overhead.
Finally, the moon is always visible with the naked eye from outdoors.
The moon appears to change form and is visible at different times of day during the month. Many people are shocked to see the moon during the day, yet it is just as visible during the day as it is at night. At a new moon, for example, the moon looks closest to the sun, rising and setting with it. At a full moon, on the other hand, the moon is high in the sky and doesn't move relative to the constellations.
The moon is brightest when it's directly opposite the sun in terms of Earth's orbit (at perigee), so that's when it shows itself most clearly. However, it can also be seen at other times of the year if the planets are aligned properly with respect to Earth. The moon disappears from view around midnight on January 1st because that's when it is on the far side of the planet from Earth. But it can sometimes be seen during the day on New Year's Day if there's a solar eclipse happening over certain parts of the world.
People have been making observations of the moon for thousands of years using everything from simple eye charts to sophisticated telescopes. Scientists use these records to learn more about the nature of our nearest neighbor with oceans and clouds and volcanoes and craters all its own.
As the moon moves away from the sun in its orbit around the Earth, more of its illuminated surface becomes visible. This is why the moon looks like a crescent or half-moon at times. During the day, it's easier to notice when it's farther from the sun and visible above the horizon. At night, when the moon is closer to the earth, it hides slightly below the horizon.
The more distant the moon is from the earth, the longer it takes for it to rotate under the horizon and disappear completely. If the moon was exactly opposite the sun with respect to the earth, then at midnight it would never rise or set because there would be no part of the moon facing towards the earth.
However, due to the fact that the moon is not exactly opposite the sun, at midnight there is a part of the moon which is facing towards the earth, allowing it to show itself again after it has gone beneath the horizon.
This is why you can still see the moon even though it is daytime. The moon is not completely hidden from view at any time of day or night, but it is simply too far away from the earth at certain points in its orbit to be visible.
The fact that the moon is not completely hidden from view at any time of day or night means that it can be seen without interfering with other things that need to be done during the day.