This is a fantastic question! I had to go out and do some research of my own to learn why the moon turns red sometimes. Most of the information below is from a video produced by NASA. I included a link to the video below. You should watch it after you read through this answer... It turns out that the moon turns red during a “total lunar eclipse”, when the moon is in Earth’s shadow.

The moon orbits around the Earth over a period of about 29.5 days (relative to the sun). Sometimes the moon passes through Earth’s shadow, causing an eclipse. This happens (on average) about twice per year. When the moon passes through the middle of Earth’s shadow, red light from the fringes around Earth beam onto the surface of the moon, turning it red. Why is there red light around the fringes of Earth when facing toward the sun? Light passes a long distance through the atmosphere in this direction, and the gases in the atmosphere let the red light pass through. Okay, maybe that sounds a little confusing… So I have included two different illustrations to help explain. The illustration 1 shows the position of the moon during a full lunar eclipse, when it’s bathed in red light. The illustration 2 explains why there is red light bent into Earth’s shadow.

This is a good transition into the second part of your question. Why does the sky appear blue? It can turn into other colors, like pink, red, and orange during sunset. Why is that? You should never look directly at the sun without the right kind of filter because the intense light can hurt your eyes. But, if you were to look directly at the sun (DON’T DO THIS) you would see that the light is white. The same is true for a light bulb in your house (you can look at this and pretend it’s the sun). White light is a combination of light of all different colors, from red to violet. Different colors have different “wavelengths”– purple and blue light has a shorter wavelength than red light. When the colors are combined, we cannot see them individually and they appear white. There are certain ways that the colors can be separated out of white light, and this happens in the atmosphere of the Earth. Gas molecules in the atmosphere “scatter” the different wavelengths of light, and it turns out that blue light is scatted the most.

We see the scattered (blue) light when we look at the sky, so it appears blue. During sunrises and sunsets, the sun is low in the sky, so light that reaches us travels a long distance through the atmosphere. The blue light is strongly absorbed by water vapor and dust and the red and pink light pass directly through (nasa.gov). I have included a picture illustrating this phenomenon and have also included a link to another informative NASA video.

video about eclipses and red moons:
video

video about the color of the sky:
video-audio click here

Thanks of the very interesting question. The reason why the moon appears red sometimes, is the same reason why the sky appears blue.

Sunlight is made of different colors of light with different intensity for each color, combined together. White light is made of all the colors in equal intensity. An interesting experiment: When white light or sunlight is passed through a prism, one can see the different colors split, because different colored light travels with different speed in certain objects such as a prism. This experiment was very important to establish the nature of light by Newton and Huygens.

It happens that our atmosphere scatters blue light more than red. This is called "Rayleigh's scattering". Therefore out of all the colors in the sunlight, blue light is the one that is scattered and reaches our eyes during the day.

During a lunar eclipse, the moon turns red. A lunar eclipse is when moon, earth and sun align in a straight line with the earth in between the sun and the moon (like so: O---o---.). When this happens, light from the sun first hits the earth. Most of the light hits the earth in the middle, but some of the light passes through the earth's atmosphere. Because the atmosphere scatters blue more than red, all the blue light is "filtered out" of the sunlight due to scattering. But red light that don't scatter remain and reaches the moon after passing through the earth's atmosphere. It is this red light that hits the moon and reflects back to our eyes, and cause the "Blood moon".

The next time you watch a beautiful sunset, observe the color of the clouds that float. They will be bright orange. This also happens when there is fire in the nearby woods. Those sunsets are very beautiful with multiple colors decorating the sky. This is because smoke scatters different shades of red light. The sky can appear redder if our atmosphere is filled with smoke.

Although, a smoke-full atmosphere will certainly be a beautiful sight to watch, it will trap the heat from the sun and raise the temperature of the earth making it uninhabitable for living beings, which of course includes us. This is called "global warming", and is an increasing problem because of the smoke from cars, coal burning, etc..

1. Light from the moon is a mixture of different colors (two examples are blue and red).
2. You can think of blue light as a wave that has a shorter wavelength than a wave from red light.

Now that we understand these, I can explain why we see a red moon. When light from the moon reaches Earth, both blue and red light bump with molecules in the air. But because blue light is shorter than red light, the molecules in the air are more able to block the blue light and scatter them away. What you're left with is the red light, which is a little more difficult for air molecules to deflect. Those red light waves are then able to pass through our atmosphere and reach our eyes.

This is why it looks like the moon has turned red. So you can see that you need a lot of molecules so more blue light can get scattered. Pollution in the air can cause the moon to turn red, because it brings up a lot of dust particles to the sky.

Now, for your second question. The sky is blue because of the same process that we just talked about. During a nice, beautiful day, what you're seeing is the blue light being scattered in all different directions by many molecules up in the sky. You're probably thinking, "well, if the moon can turn red, can the sky do that too?” Yes, it actually can. That's what happens if you've ever watched a sunset. The sky turns reddish and yellowish due to the same process that makes the moon turn red. When the sun is low in the horizon (just like what happens during a sunset), light from the sun (which is also a mixture of different colors) gets blocked and scattered by the molecules in the air. So the sky turns red.

I realized this is a really long answer, but you have very interesting questions. Good luck with your studies!

The sky is blue because blue light scatters off of air molecules more than red light does. Violet light scatters even more, but so little of the sun's light is violet that we see the sky as being blue.

The moon appears red during a lunar eclipse because it is in the shadow of the Earth. I do not know whether the red color is filtered through dust in the Earth's atmosphere as light bends around the planet, or if it is reflected light from the night side of the Earth.

Light is a very interesting phenomenon. Light that is visible to our eyes can have all the colors that you see in a rainbow. If light with two different colors gets mixed together, it will become a third color. For example, green light and red light taken together becomes yellow, blue light and red light becomes magenta (close to pink) and so on. The light that comes from the sun contains all the colors together and therefore appears white.

However as light travels in a straight line from the sun into your eyes it needs to get through the atmosphere. Small particles or gas in the atmosphere “take out” a lot of the blue light and only very little of the red light. Thus, when you look at the sun, a lot of the blue light has been taken out by particles in the atmosphere and the sun looks yellow (Do not look at the sun without the right equipment. It will hurt your eyes very much!). The particles in the atmosphere cannot hold on to the blue light forever so they throw it back out in random direction. We say that the light gets “scattered”. Some of this light gets scattered back into space and is lost and some of the light reaches your eye. Thus, when you look at the sky, what you see is the blue light that was “taken out” from the sunlight by particles in the atmosphere and was then scattered around and reached your eye.

At the beginning or the end of the day, when the sun is very low on the horizon, the path of the light through the atmosphere is very long and a lot of the blue light gets lost along. Therefore, the sky will appear red and orange. The same is true when the atmosphere is very “dirty” for example because of pollution from cars or when a volcano erupts and spits ash into the air. Again, the particles scatter off the blue light and when enough of it gets lost, all that is left is the red light.

Now you can understand why the moon can be red. The moon just reflects the white light from the sun and sends it to the earth.

So similarly to the sun, when the moon is low in the sky, the light has to travel a long way through the atmosphere and a lot of the blue light gets lost. The moon appears red. The other way to get a red moon is when there are a lot of particles in the air due to pollution.

Finally, the moon can be red when the sun is on the other side on the earth and positioned so that the moon does not receive any direct white light from the sun any more. The only light it receives is the red light that made it through the atmosphere of the earth. Therefore the moon now reflects red light back to the earth and appears red.