The Hiroshima bombing of 1945, as you may know, resulted in the destruction of most of the city. As a result the atmosphere over Hiroshima at the time must have been polluted with great amounts of smoke from the bomb explosion, as well as soot and other particles from the dust and rubble caused by the explosion. When the rain came down it carried with it these impurities and therefore must have appeared black. This is similar to 'acid rain' which refers to the slightly acidic rain that falls in highly polluted areas, due to the dissolution of atmospheric carbon dioxide, sulfur dioxide and nitrogen oxides. 'Blackrain' was reported by many eyewitnesses in the aftermath of the Hiroshima bombing and has been used commonly in the media to refer to the aftermath in general.

When a large explosion occurs, a giant plume or up draft of air is produced. This up draft is the STEM of the MUSHROOM cloud. Now this updraft carries with it vaporized material as well as particulate matter (dirt and junk from the target) up very high about 50-60 km into the atmosphere. When it rains, this black sooty stuff comes down giving a black rain.

I had never heard this, so I did some research on line. The first web site to come up was a BBC (British Broadcasting Corporation) interview with a Hiroshima survivor, Keiko Ogura that appeared on the radio on August 5, 2005. This is what she had to say about the black rain:

"There was black rain falling, black rain mingling with ashes and rubbish and oil, something like that. It smelled bad and there were many spots on my white blouse - sticky, dirty rain."

black_rain_spots

I found another question similar to yours that was answered on a web site sponsored by the Singapore Science Center. The answerer said that when an atomic bomb is dropped and hits the earth, a large fireball develops close to the ground. The warm air in the center of the fireball rises, causing a massive wind to blow toward the base of the fireball, to replace the air that rose above it. This wind draws in dirt and dust as well as ash and debris from the initial bomb blast. As the dirt, dust, ash and debris become mixed with radioactive fallout from the bomb; the material becomes coated with radioactive particles, and becomes radioactive itself. This dirt, dust, ash and debris remain in the atmosphere until it rains, when the material mixes with rain drops and falls back down as black rain, coating everything with radioactive dirt for hundreds and even thousands of miles from the bomb site. I'm sure a similar thing happens, on a smaller scale, when any large bomb or meteorite hits the Earth (except that radioactive black rain will only fall when the bomb is radioactive, as it was at Hiroshima).

There's also a book about the Hiroshima bombing, based on interviews with survivors, entitled "Black Rain: A Novel", written by Masuji Ibuse and translated from the Japanese by John Bester, so you might want to check that out. It appears that there is a movie of the same title as well, filmed in black and white in Japan in 1989. It seems that black rain has become a metaphor for the atomic bomb itself in Japan.