There are state schools in the majority. Some schools are in ecclesiastical sponsorship, but awarded the state-recognized degrees. There are some private schools that charge different levels of school fees.
In Hamburg, there are two types of secondary schools, the district schools and high schools/gymnasium.
District School
The district school covers the grades 5 to 13 and offers all school graduations up to graduation from high school. In the district school all pupils learn together. Children who are weaker in their performance, as well as more capable ones, are specifically supported and demanded. The district school is more focused on the acquisition of competences for the design of the professional lifepath (vocational and study orientation).
http://www.hamburg.de/stadtteilschule/2670728/stadtteilschule/
Grammar School/Gymnasium
The grammar school covers the grades 5 to 12. It teaches pupils with the goal to lead them after grade 12 to the general university entrance qualification. The general grammar schools offer high-performance pupils not only subject teaching but also interdisciplinary teaching, opportunities for specialization in terms of performance and inclination through numerous choices in languages, natural sciences and the aesthetic disciplines and support for particularly gifted pupils (promotion of excellence). At grammar school all school graduations can be made.
http://www.hamburg.de/gymnasium/
Information about the vocational schools you will find in the age group 16-18.