Originally, the term public school in the United Kingdom was used to describe just seven schools: Charterhouse School, Eton College, Harrow School, Rugby School, Shrewsbury School, Westminster School and Winchester College. Today, however, the term may be used for any U.K. independent (fee-paying) senior school which belongs to the association known as The Headmasters and Headmistresses Conference (HMC). Confusingly, in the U.K., public schools are sometimes referred to as private schools!
Oxbridge is a portmanteau word formed from Oxford and Cambridge. As a noun, it is used to describe the Universities of Oxford and Cambridge as an entity. It may also be used as an adjective to describe the attributes (or perceived attributes) of the two universities: as in, for example, 'he spoke with an Oxbridge accent'.
The suggested link between public school/Oxbridge education and Conservative politicians is borne out by historical fact. Eleven Tory/Conservative British prime ministers were educated at Eton and Oxbridge. These include Robert Gascoyne-Cecil (Eton/Christ Church, Oxford); Arthur Balfour (Eton/Trinity College, Cambridge); Sir Anthony Eden (Eton/ Christ Church, Oxford), Harold Macmillan (Eton/Balliol College, Oxford) and David Cameron (Eton/Brasenose College, Oxford). Having said that, 8 former pupils of Eton went on to become either Whig or Liberal prime ministers. These include William Ewart Gladstone (Eton/Christ Church, Oxford) and Archibald Primrose, 5th Earl of Roseberry (Eton/Christ Church, Oxford).
It is noted on this website that of the 55 British prime ministers to date, 41 were educated at Oxbridge, although it should be noted that not all Oxbridge-educated prime ministers attended public schools.
(Please note that there are some variations in statistical information about British Prime Ministers on the Internet).