Many cultures (though not all, see below) feature children's songs and verses that are passed down by oral tradition from one generation to the next, however the term "nursery rhyme" generally refers to those of European origin. The best known examples are English and originated in or since the 17th century. Some however are substantially older, "Baa Baa Black Sheep" exists in written records as far back the Middle Ages. Arguably the most famous collection is that of Mother Goose. Some well known nursery rhymes originated in America, such as "Mary had a little lamb".
Generally nursery rhymes are innocent doggerel, though some scholars have attempted to link their meaning to events in European or English history. Urban legends abound with regard to some of the rhymes, though most of these have been discredited. Some of the more plausible explanations indicate that some rhymes may have been contemporary social or political satire. ("Hey Diddle Diddle" is one example, the "dish" and "spoon" possibly being nicknames for the figures involved in a sex scandal in the court of Elizabeth I.)
"Ring-Around-the-Rosie" (alternatively "Ring-a-ring of Rosies") is popularly believed to be a metaphorical reference to the Great Plague, although this has been widely discredited, particularly as none of the "symptoms" described by the poem even remotely correlate to those of the Bubonic plague, and the first record of the rhyme's existence was not until 1790.
A credible interpretation of "Pop goes the Weasel" is that it is about silk weavers taking their shuttle or bobbin (known as a "weasel"), to a pawnbrokers to obtain money for drinking. It is possible that the "eagle" mentioned in the song's third verse refers to The Eagle freehold pub along Shepherdess Walk in London, which was established as a music hall in 1825 and was rebuilt as a public house in 1901. This public house bears a plaque with this interpretation of the nursery rhyme and the pub's history. Alternatively, the term "weasel" might be Cockney rhyming slang for a coat ("weasel and stoat" = "coat"), and the coat itself was pawned.
Scholars occasionally think they have "all" nursery rhymes written down, or know the last time that a rhyme was in use (some fall out of favor). However, due to the fact that they're mainly an oral tradition, nursery rhymes will "pop up" anew. See Bill Bryson's book "Made in America : An Informal History of the English Language in the United States" for an excellent example.
There are some aboriginal tribes which consider music sacred, so that only elder men may sing songs, and the songs are taught during sacred rituals in adulthood. It is forbidden for women or children to sing. Hence, these cultures don't have these kinds of songs.
List of nursery rhymes