Usually red color of moon is seen during an eclipse. The reason behind this is exactly the same that why sun appears red during sunrise and sunset. Sun always shines, when there is night in one part of the earth the other part is having day. When the earth comes between the sun and the moon sun rays fall on earth which causes shadow on the moon. The sun, the moon, the earth is in such a position that sunlight has to pass through the atmosphere of earth to reach moon. This happens only during the eclipse.
So the earth atmosphere scatters the light coming from the sun but longest wavelength of red color passes through the atmosphere as it is causing the red colored shadow on the moon. Same is the reason why sun appears reddish during sunset and sunrise because at that time the light has to travel more atmospheric distance.
Change in color also happens when very large volcanoes erupt which releases a huge amount of dust particles and smoke in the atmosphere. Some amount of change is observed due to the burning of forest. This make stars and other very far away objects in the universe appear not very clear to the human eye. If there was no atmosphere on the earth it has casts black shadow instead of red and we were not able to see the moon at all during the eclipse.