We've used the expression Blue Moon simply to mean the second Full Moon in a single calendar month. The Moon doesn't actually change colour, despite the name.
However, there have been occasions when the Moon has appeared to be blue in hue. This isn't an astronomical phenomenon. Instead, it is caused by dust or smoke high in the Earth's atmosphere. The dust is thrown up by major volcanic eruptions such as Krakatoa, Mount St. Helens or Mount Pinatubo, whilst smoke can come from large forest fires.
Whether it's dust or smoke, the tiny particles have a strange effect on the moonlight (or sunlight) passing through them. They scatter the light in every direction, but red light is scattered more strongly than blue light, so that less red light passes directly through the dust or smoke. Thus the Moon has a blue tinge.
Because of the unusual size distribution of the dust particles, this scattering works in the opposite sense to the normal scattering by dust low on the horizon which makes the Sun appear deep red in colour as it rises or sets.