In the UK, there are various kinds of schools:

Arranged by level:

Nursery Schools - Ages about 4.
Primary Schools - Ages 5-11
First Schools - Ages 5-8
Middle Schools - Ages 8-13
Secondary Schools - Ages 11-18
Upper Schools - Ages 13-18
Sixth Form Colleges - Ages 16-18

Not everyone gets all of these. There are two main transitions:

Primary -> Secondary
Lower -> Middle -> Upper

Many Primary schools are divided into Junior and Infant departments - Junior being the Senior group.

Ages 18+ are covered by various Higher/Further Education systems, which we do not tend to refer to as schools in the way which Americans do. At that level, school denotes a faculty.

Arranged by organisation:

State Schools: The standard. Provided by the government.
Prep Schools: Short for 'preparatory', these provide courses suitable for those entering public or private schools, and run from 7 to 11 or 13.
Private Schools: Privately-run schools that typically cost money to attend.
Public Schools: Private secondary- or upper-level schools of the highest quality. Expensive, elitist, but effective.

Grammar Schools: Now almost obsolete. State-run secondary schools with entrance examinations. Abolished, as they were considered elitist. They provided a better level of education, free, to anyone who passed the exam. Nowadays, in the interests of equality, this level of education is only available to those willing and able to pay. A few remain, after a fashion, in some parts of the country.
Secondary Modern Schools: Where you went if you failed to get into a grammar school. Under-financed, poorly-thought-of, not missed.
Comprehensive Schools: The replacement for the former two: much more like the latter than the former, generally. Referred to be the present (Labour) government as 'bog standard'.

Thanks to everyone for their additions.

Addendum: SpudTater gives some insight into the 'form' notation, and I'll attempt to add some more.

Schools traditionally numbered their year-groups from youngest to oldest, starting at '1'. In secondary, upper, and preparatory schools, the years were called 'forms', and the last two years (ages 16-18) were collectively the 'sixth form', divided into 'lower sixth' and 'upper sixth'.

In recent years, a standardised state system has been introduced, running from 'Reception' or year 0, at the bottom of the infant schools, to 'Year 13' - the old Upper Sixth. Many people find the zero-indexing of the new system confusing. The zero-year is often referred to as 'R', just as the National Curriculum zero attainment level is referred to by letter, so that no-one actuall registers a zero achievement.