Some people believe that this rhyme refers to the Black Death that spread through much of Europe during medieval times.

"Ring around the rosy"
Those suffering from the plague would get terrible boils which would be encircled by a dark ring of dead skin.
"A pocket full of posies"
The doctors and physicians of the day would carry flowers in their pockets to mask the terrible smell that the people with the plague emitted.
"Ashes, ashes"
Again referencing the dark skin color that is one of the plague's symptoms.
"We all fall down."
This line equates "falling down" to dying."

Many folklore scholars completely disregard the comparison of the nursery rhyme to the plague for several reasons including the fact that many of the "symptoms" described in the rhyme were not actual symptoms of the Plague, and some early translations of the rhyme imply that the "falling down" was actually a polite bow to one's dancing partner.

The true origin is from the mid-to-late 1790s in England, but I think the Plague speculations are far more interesting.

The words to the Ring around the rosy children's ring game have their origin in English history . The historical period dates back to the Great Plague of London in 1665 (bubonic plague) or even before when the first outbreak of the Plague hit England in the 1300's. The symptoms of the plague included a rosy red rash in the shape of a ring on the skin (Ring around the rosy). Pockets and pouches were filled with sweet smelling herbs ( or posies) which were carried due to the belief that the disease was transmitted by bad smells. The term "Ashes Ashes" refers to the cremation of the dead bodies! The death rate was over 60% and the plague was only halted by the Great Fire of London in 1666 which killed the rats which carried the disease which was transmitting via water sources. The English version of "Ring around the rosy" replaces Ashes with (A-tishoo, A-tishoo) as violent sneezing was another symptom of the disease. The first printing of the rhyme was in Kate Greenaway’s 1881 edition of Mother Goose. The rhyme must already have been widely distributed. A novel of 1855, The Old Homestead by Ann S. Stephens, shows children playing "Ring, ring a rosy" in New York.5 William Newell reports two versions in America at much the same time as Greenaway (1883) and says that another was known in New Bedford, Massachusetts around 1790. But the relation of this rhyme to the plague is considered to be nonsense because this was not a theory until 1961. There really is no true meaning of the song. It has an indefinite origin and no specific meaning and dates back to 1347.

http://www.rhymes.org.uk/ring_around_the_rosy.htm

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http://www.snopes.com/language/literary/rosie.asp