In the UK:
Public school means fee paying aka private.
State school is the equivalent of local school here - if you call it 'public school' people will think you mean private or fee paying.
Grammar schools are a bit of an old fashioned educational institution that I believe one of the Labor governments wanted to get rid of (back in the 60s or 70s) but they've got a good reputation and they stayed. They seem to be fairly few and far between but exist in little clusters in various parts of the country. I went to a grammar school and it is constantly graded well by Ofsted and in my opinion had many good features that would rival most of the international schools here.
To get into grammar school aged 10/11, a child has to take the 11+ exam - I'm not sure what they do if they're older and want to go - perhaps they also have to take some kind of test, or the school may base it on reports from their old school. If they fail the 11+ the option is to go to the local comprehensive or secondary school instead.
There are good and bad points to both kinds of school in my opinion. Although my grammar school was a very good school, it's aim was for most students to go to university. A student may be smart enough to pass the 11+ and get in there, but that doesn't necessarily mean that by the time they get to 16 they will still want to go to university...........this happened to myself and quite a few of my friends and in that case, the provision given to these non-university students wasn't too good - not a lot of career advice there. However, this was back in the 80s and I'm sure they're more understanding nowadays of those who aren't interested in studying in 6th form (as it is often called in the UK............people here would know it as Grades 11 and 12, or perhaps years 12 and 13, depending on which school their children are at).
Friends of mine who went to the local comprehensive school (ie those who failed the 11+) found that the school was very geared up to those leaving at 16 and offered good trade-based skills as part of the curriculum - ie metalworking/woodworking/childcare etc - they also sent their students on work experience when they were 16.......something that our school didn't do (I did that at college instead). This particular comprehensive school now has a good 6th form so the students have the best of both worlds there - they can get trained up for leaving at 16 or they can stay on and do A'levels.
Anyhow, as your child is still young, all this will not apply immediately but it's worth considering for the future. When looking at primary schools in the area, check out the secondary schools too.
There's a website - I think it's called the Good Schools Guide. You have to pay to use the site but it's not too much - a couple of pounds per month, so that'd be worth doing. I believe you can also find out about things like catchment areas on that site. There's also Ofsted.
You may find the housing website 'rightmove' useful as you can view houses to buy or rent and they have a map feature which will show the schools in the area with a link that takes you directly to the Ofsted reports for those schools.
I hope all of this rambling is of some help to you
