Before the grammar, it helps to understand the German university system itself, since its vocabulary does not always map neatly onto other national systems, and confusing the two causes real misunderstandings when enrolling or describing your academic path. German universities are structured in three main stages: Bachelor (typically three years or six semesters), then Master (typically two years), then Promotion or Doktorarbeit for a doctorate. This structure comes from the European Bologna Process and resembles other three-cycle systems in shape, but uses its own German terms that must simply be learned as they are.
There is also a fine distinction between Universität and Hochschule: the former is a traditional research university granting academic, theoretical degrees, while the latter — often a Fachhochschule — leans toward applied, technical training. Choosing the right word when describing your path gives an accurate impression of what kind of education you have.
Inside the classroom, terms divide by the nature of the session: die Vorlesung is a large lecture where students mostly listen without much interaction; das Seminar is a smaller class requiring participation, discussion, and sometimes a presentation (Referat); die Übung is a practice session for working through exercises tied to the lecture. Each stage of study ends in its own written project: a Hausarbeit is a term paper written during the semester, while a Bachelorarbeit or Masterarbeit is the final degree thesis.
Grammatically, note that the core university verbs impose fixed constructions that must be memorized precisely: studieren takes the subject of study directly in the accusative with no preposition (Ich studiere Informatik, not in Informatik), while sich einschreiben an requires the preposition an with the dative, since it points to the place of enrollment (the university). This distinction between a verb taking a direct object and one requiring a preposition is where many learners slip, often inserting a preposition after studieren by analogy with phrases like "study in" in their own language — but German needs no preposition here at all.