uk public schools are public in the sense if you have the money for tuition you can go, wheras "private" schools are only open to invited families, no matter how wealthy they are
Public schools - a handful of top level very expensive private schools, usually have existed for centuries, the children of the highest levels of society tend to go there, a great foundation for a career in public life. Examples include Eton, Harrow, Winchester, Marlborough.
Private schools - the rest of the private schools, less expensive, but less advantageous.
State schools - free, provided by the state for children in a local catchment area, this is where the rest of us go. Most are run by the local government, but there has been some experiments with 'foundation schools' which are tax payer funded but privately run.
There are also a handful of true grammar schools left. These are free to attend, but are selective based on ability. There is debate about the extent to which raw ability is really being tested, and not preparedness for the test.
There are also private schools that call themselves grammar schools. These were once free and selective, but converted to fee-paying at some point. They like the prestige of being called a "grammar" but don't be fooled!
The lack of true grammar schools has been a big hit to social mobility in the UK. Now, getting into a grammar requires not just passing a test, but also having parents rich enough to buy a house in the catchment area. Public school alumni dominate top industries and government.
The UK "free school" approach has been a mixed bag. On the one hand, various academy chains have managed to turn around failing schools. On the other, some parents are left with no choice but to send their child to a religious school due to catchment areas and waiting lists.
As a tax payer, I don't want my contributions going towards religious institutions.
The loss of "special schools" for students with severe special needs has not helped anyone (imo).
You see a few parents suddenly get religion as their child turns 9 or 10 and they realise they are unlikely to get into a grammar school.
If you pass a specific test (the 11+) you can go to a selective grammar school.
There also used to be government funded scholarships to private schools for the very brightest underprivileged children but that was stopped by Labour, the "party of the Working Class".
That's not to say that all things being equal you wouldn't stand slightly more chance if you were a member of the aristocracy.
https://en.wikipedia.org/wiki/Public_school_(United_Kingdom)