Public schools are different things on either side of the Atlantic Ocean. Both sets of schools are heavily influenced by the Prussian schooling model. Introduced in the early 19th century, the Prussian model mandated many appalling ideas, including:
- Education should be compulsory
- Teachers should be trained in teaching (shock! horror!)
- Exams should be standardised so that exam results should be comparable
- There should be a national curriculum so that all pupils learn the same things
- Mandatory kindergarten
 American public schools
Public schools in the United States are schools paid for by government funds to educate all students in the area for free or minimal cost. People with an axe to grind call them "government schools," which is something of a shibboleth for conservative libertarians. Because of the First Amendment to the US Constitution, public schools in the US are not allowed to hold religious functions, which tends to get Christian underwear in a twist. Also they tend to want to teach evolution in science class. Apparently that's because evolution is science or something, but this time the twisted underwear mainly belongs to fundamentalists because mainstream Christians don't have such a problem with it. Some radical morons feel that the Department of Education itself is unconstitutional.
 British public schools
What are known as public schools in America and other countries are called state schools or comprehensive schools the UK. They are funded by the government and attended by the vast majority of British children.
The term "public schools" is used in Britain to refer to a small number of exclusive (and normally expensive, though there are scholarships) private schools, the most famous being Eton, Harrow, Rugby, Stonyhurst, Ampleforth and St. Custards. Exactly what is public about them is a bit of a mystery to someone who didn't grow up with the term, but the name is derived from the time when anyone (the public) could attend, given the cash necessary - most schools at the the time of their foundation requiring involvement in the Church before one's son would be admitted.
Traditionally British public schools were boarding schools where parents from the elite would send their sons to get beaten by headmasters, housemasters and prefects. This was believed to be "character-building." Today corporal punishment is illegal in British schools. Public schools often have esoteric traditions dating back several generations; one of the most famous public school traditions, now discontinued, was "fagging" (nothing to do with homosexuality, although reputedly a lot of that goes on at public schools too). Younger pupils used to be selected as "fags", acting as servants to school prefects (an elite group of older students). Until the mid-twentieth century, prefects had the right to cane or otherwise discipline their fags and other younger pupils. The Duke of Wellington is often quoted as saying that "The Battle of Waterloo was won on the playing-fields of Eton," although it is unlikely that he did, particularly as it didn't actually have any playing fields at the time.
Nowadays public schools are generally regarded as rather elitist, giving rise to the "old school tie" environment in "the City," As any fule kno, "the City" refers to the financial and similar companies/organisations primarily in the City of London (The "Square Mile" - not to be confused with Greater London). Rich parents pay very high fees to get their children (usually sons) a select education at public schools. There are also public schools for girls (Roedean being the most prestigious) and some boys' public schools accept girls at the senior level in the 6th form. The most successful public school students conventionally progress to Oxford and Cambridge Universities (Oxbridge). Today there are a high number of state school-educated students at these universities, but there is still a disproportionately large public school contingent.