Humanistisches Gymnasium (humanities-oriented)
Humanities-oriented gymnasia usually have a long tradition. They teach Latin and Ancient Greek (sometimes also Old Hebrew) and additionally teach English or French or both. The focus is on the classical antiquity and the civilizations of ancient Greece and ancient Rome.
For certain subjects, such as History, many universities still require the Latinum, some also the Graecum, proof of study or comprehension of Latin or Ancient Greek, respectively.
Neusprachliches Gymnasium (focus on modern languages)
This type of school is less traditional. It teaches at least two modern languages. In most cases the students have the chance to learn Latin as well.
Mathematisch-Naturwissenschaftliches Gymnasium (focus on math and science)
Often combined with the Neusprachliches Gymnasium this type of schools have a focus on STEM subjects.
Previous names
The Gymnasium with focus on modern languages used to be called Realgymnasium, the Gymnasium with focus on math and sciences used to be called Oberrealschule. The Gymnasium was supposed to be the humanities-oriented variety; a common term for all of these schools put together was Oberschule (literally, "upper school"). In the 1960s, school reformers in an equalization effort discontinued these names.