The Moon makes a full orbit around the Earth once per month. From full moon to full moon, it is 29.53058886 days (synodic month).
Full moon happens when the Moon is at one side of the Earth (right side in the image), and the Sun is at the other side. This way, Moon is fully iluminated by Sun (except in the cases where it is exactly in the opposite side of the same line, where you have a Moon eclipse). In this phase, Moon is at its highest altitude at midnight.
Then, as the Moon orbits, it comes closer to the Sun in the sky. This means both that it shows us less iluminated percentage and that its rise and set get delayed. At Last Quarter, Moon is at its highest altitude at sunrise. This means that you will be able to see the half moon figure on the sky during the morning (and last half of the night).
Then it comes New Moon, where the Moon is nearest to the Sun in the sky, and almost invisible, since it shows us the dark side during the day. Its highest altitude is at noon, but you will not be able to see it. You can, if you are lucky, see a Sun Eclipse.
Finally it comes First Quarter. To put it short, highest altitude is at sunset and you are able to see Moon during all the afternoon (and first half of the night).
Answer from Envite on Stack ExchangeELI5: How come sometimes we can see the Moon during daytime?
Why can we sometimes see the moon during the day? - Astronomy Stack Exchange
Why can we see the moon during daytime?
Why can we sometimes see the moon during the day and what do people who are in nighttime on the other side of the world see?
Can I see the new moon during the day?
How often is the moon up during the day?
Why doesn't the full moon appear during the day?
The Moon makes a full orbit around the Earth once per month. From full moon to full moon, it is 29.53058886 days (synodic month).
Full moon happens when the Moon is at one side of the Earth (right side in the image), and the Sun is at the other side. This way, Moon is fully iluminated by Sun (except in the cases where it is exactly in the opposite side of the same line, where you have a Moon eclipse). In this phase, Moon is at its highest altitude at midnight.
Then, as the Moon orbits, it comes closer to the Sun in the sky. This means both that it shows us less iluminated percentage and that its rise and set get delayed. At Last Quarter, Moon is at its highest altitude at sunrise. This means that you will be able to see the half moon figure on the sky during the morning (and last half of the night).
Then it comes New Moon, where the Moon is nearest to the Sun in the sky, and almost invisible, since it shows us the dark side during the day. Its highest altitude is at noon, but you will not be able to see it. You can, if you are lucky, see a Sun Eclipse.
Finally it comes First Quarter. To put it short, highest altitude is at sunset and you are able to see Moon during all the afternoon (and first half of the night).
Well, the Moon reflects quite a bit of the Sun's radiation. In fact, this person states that:
The visual geometric albedo of the full moon is 12.5%, but much less at other phases.
In combination with this, the Moon is also the closest astronomical object to us, and since the intensity of light falls off as 
(inverse-square law), this plays arguably the biggest role in being able to see the Moon during the day. In fact, so much light from the Moon gets reflected to Earth that its apparent magnitude must be larger than the sky during the day (otherwise it would get washed out by sky noise).
According to this article, the apparent magnitude of the daytime sky is somewhere between 1.5 and 3. When you compare this to the full moon (–12.74; though you really can't see this during the day due to the geometry of the Earth/Sun/Moon system), or even a new moon at its brightest (-2.50), the Moon can almost always be seen in the sky.
It seems weird to me.