The equivalent f the US "private school" is the UK "independent school". Within the umbrella term "independent school" (also referred to as "fee-paying school" in tabloids to emphasise the socioeconomic separation), there are both public schools and private schools.

The distinction between the two arose in 1868 with the passing of the Public Schools Act, which was the UK government's way of "rewarding" the most important/influential schools of the day by loosening their grip on them. Effectively, the U.K. Government of the day realised that by imposing a particular curriculum on the highest achieving schools, they were restricting the learning opportunities of students, so they kind of "devolved" some of that authority to the schools.

Two schools are technically independent and public but not usually referred to as public schools, St Paul's and Merchant Taylor's (I believe). This strange situation stems from the following exchange:

UK Gov't: Hey good schools, you know how we told you what you had to do before? Well, we trust you enough to set your own curriculum now, you're free to do as you wish!

St Paul's: Good one mate, you never had the right to tell us what to do, we have pretty much always set our own curriculum; don't you dare include us in that list and make it seem as though you had any say in the first place, you dicks.

Merchant Taylor's: Yeah! That!

U.K. Gov't: Well, okay then, you're exempt.

So now you know what public schools are! Other fee-paying independent schools which were neither involved in discussions about or subject to the Public Schools Act can be referred to as "private school", so the separation becomes "State", "Private", "Public", "Misc" (for SPS and the other aforementioned school).

Within those, there are obviously subcategories (state schools exist as "academies", as normal schools, as selective, independent-like schools supported by tax money, like Tiffin for example, etc), but those are the major subdivisions.