The examination is called the eleven plus. Partly due to the failure to fully implement the tri-partite system prescribed by the 1944 Education Act, the examination came to be seen as delivering a pass/fail result with the academically selected pupils passing and attending grammar schools and the remaining pupils being deemed to have failed and being consigned to schools euphemistically designated Secondary Modern Schools.
This arrangement proved politically unsustainable, and, over the period 1960 to 1975, non-selective ("comprehensive") education was instituted across a substantial majority of the country. The eleven plus examination had been championed by the educational psychologist Cyril Burt and the uncovering of his fraudulent research played a minor part in accelerating this process.
To understand grammar schools in the UK, some history is needed. After WW2, the government reorganised the secondary schools into two basic types. Secondary moderns were intended for children who would be going into a trade and concentrated on the basics plus practical skills; grammar schools were intended for children who would be going on to higher education and concentrated on the classics, science, etc. This system lasted until the 1960s, at which point changes in the political climate led to the general acceptance that this was a discriminatory system which was not getting the best out of all children. This was partly because some authorities tended to prioritise their budgets on the grammar schools, damaging the education prospects of children attending secondary moderns.
The decision was taken to switch to a single type of school designed to give every child a complete education. That is why this new type of school is called a comprehensive school. However the timetable of the changeover was left to the local authorities, some of whom were very resistant to the whole idea and thus dragged their feet for as long as possible. The result is that there is now a mixture. Most authorities run a proper comprehensive system, a few run essentially the old system of secondary moderns and grammar schools (except the secondary moderns are now called "comprehensives"). Some run comprehensive schools along side one or two remaining grammar schools.
The Labour government that came to power in 1997 instituted measures that allowed parents to force a local referendum on whether to abolish grammar schools in their area. The form of this referendum depends on whether there is still a full two-tier system running, in which case all parents with children at primary schools in the area are eligible to vote, or whether there are only a few grammar schools in the area, in which case only those parents with children at primaries that regularly send children to the grammar school are eligible. By 2003, only a few referenda had taken place and none of these had delivered the requisite majority for conversion.
The debate over selective education has been widened by other measured introduced by the Labour government, allowing schools to select a portion of their intake by 'aptitude' for a specific subject. There are many who think that selection allows children to receive the form of education best suited for their abilities, while "one-size-fits-all" comprehensives fail everybody equally. One of the greatest attacks on the comprehensive system is that it leads, in essence, to selection on the grounds of wealth as the good schools are generally located in areas with expensive housing, so children from poor areas are denied the possibility of attending them. Conversely, there are many who think that the selection of children at 11 divides them into 'successes' and 'failures' at that age, and is therefore wrong. The current Labour government, from the party that originally championed comprehensive education, appears to favour the first of these groups, and their introduction of local referenda on grammar schools has been attacked by opponents of selective education as an unworkable system designed to give the semblance of choice while maintaining the status quo.
Private schools generally give the same sort of education as grammar schools, but there are exceptions, Gordonstoun for one. In areas where the local authority provides a comprehensive education -- which some parents don't like for various reasons -- independent schools are particularly common.