Sports Say at St Mohammed's was always popular with the boys
Rather a misnomer nowadays, as the term 'public' implies that they're open to any old chav who fancies a spot of education. Not so. One's parents must be sufficiently solvent as to afford the outrageous fees demanded by such schools, but at least some sort of schooling is guaranteed. This is in direct contrast to what is offered by the state system - which is fuck all, unless you wish your kids to be brainwashed in to homosexuality by the time they're reached puberty and to 'graduate' with little more than a GCSE in African dance.
The term 'public school' comes from the time when the state didn't provide education [So what's new?]. The term distinguished between education in a school generally provided by a church and (thus) open to 'public' applicants, and mentoring in a 'private' household - as was then the practice within the Upper Class.
Public schools are very popular amongst the political elite - most of whom are ex-public school. Typically, these are the cnuts who extol the virtues of the state system - the failure of which they are largely responsible for - whilst sending THEIR kids to... wait for it... public schools.
Very few university students come from the state system, despite the government trying to force their entry by cajoling colleges to accept mong üntermensch with abysmally low intelligence but with (oddly) brilliant exam scores.