1. What is a grammar school?
In England, people usually mean a state-funded secondary school that selects pupils by academic ability, most often through an entrance test in Year 5 or Year 6 (commonly called the 11+). If your child meets the school's threshold and ranks high enough under its admissions criteria, they can be offered a place—without fees, like other state schools.
Grammar schools are not the same as private (independent) schools, which charge fees and follow a different admissions regime. They are also different from partially selective schools that only select a slice of their intake: those exist too, but "grammar school" normally implies a fully selective or predominantly selective admissions model.
The word "grammar" is historical. Today it signals academic selection, not a particular teaching style—though many grammars are high-attaining and academically oriented by intake.