das Halbjahr
half-year
A period of six months.
Im ersten Halbjahr des Jahres ist der Umsatz stark gestiegen.
In the first half of the year, revenue increased sharply.

half-year / semester
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half-year
A period of six months.
Im ersten Halbjahr des Jahres ist der Umsatz stark gestiegen.
In the first half of the year, revenue increased sharply.

'Halbjahr' (half-year) is an easy-to-understand compound made up of 'halb' (half) and 'Jahr' (year). So it literally means half a year or six months.
Das Projekt soll im nächsten Halbjahr abgeschlossen werden.
The project is supposed to be completed in the next half-year.
The context tells you whether a general period of time or a school semester is meant. Words like 'Schule' (school), 'Universität' (university), 'Noten' (grades), or 'Zeugnis' (report card) point to an educational context. Words like 'Umsatz' (revenue), 'Wirtschaft' (economy/business), or 'Bericht' (report) point to a business context.
Im zweiten Halbjahr müssen die Schüler eine wichtige Prüfung schreiben.
In the second semester, the students have to take an important exam.
The word is very often used in combination with 'erstes' (first) or 'zweites' (second) to refer to the first or second half of a year.
Die meisten Buchungen erhalten wir im zweiten Halbjahr.
We receive most bookings in the second half of the year.
semester
'Semester' and 'Halbjahr' can both mean a period of about six months. However, 'Semester' is more specific and mainly belongs to college or university contexts. 'Halbjahr' is more general: it can simply mean six months, or a section of a school or academic year. That is why the words can sometimes be used similarly in a study context. But 'Semester' is more natural when you are talking about lectures, exams, and university dates. 'Halbjahr' fits better when you are speaking generally about six months or about school.

Im ersten Halbjahr hat Paul viel Geld für seine Reise gespart.
In the first half of the year, Paul saved a lot of money for his trip.
Meine Freundin schreibt sich vor dem nächsten Semester für einen Sprachkurs ein.
My friend is enrolling in a language course before the next semester.