das Fach
subject
A specific field of knowledge or topic that is taught at school or university.
Mathematik war schon immer mein schwächstes Fach.
Mathematics was always my weakest subject.

subject / compartment
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subject
A specific field of knowledge or topic that is taught at school or university.
Mathematik war schon immer mein schwächstes Fach.
Mathematics was always my weakest subject.

'Fach' is a very productive base word for many compounds. If you see '-fach' at the end of a word, it often refers to one of the meanings, e.g. 'Schulfach' (school subject), 'Lieblingsfach' (favorite subject), 'Handschuhfach' (glove compartment), or 'Postfach' (mailbox/post office box).
Welches war dein Lieblingsfach in der Schule?
What was your favorite subject at school?
You can easily recognize the meaning of 'Fach' from context. If you are talking about school or university, it means a subject. If it is about a cupboard or a shelf, it means a compartment. In the context of work and expertise, it means a specialty.
In welchem Fach schreibst du morgen eine Klausur?
In which subject do you have a test tomorrow?
In the context of school or university, 'Fach' is often combined with verbs like 'belegen' (to take/enroll in), 'wählen' (to choose), 'studieren' (to study), or 'unterrichten' (to teach).
Im nächsten Semester möchte ich ein neues Fach belegen.
Next semester, I would like to take a new subject.
The idiom 'vom Fach sein' means that someone is an expert in a particular field. It is often used to express competence.
Frag am besten den Elektriker, der ist vom Fach.
It's best to ask the electrician; he's from the field.
This meaning of 'Fach' is a technical term from the world of theater and opera. It describes a certain kind of role for which a performer is especially suited, e.g. 'das komische Fach' (comic role type).
Die Sopranistin wechselte vom lyrischen ins dramatische Fach.
The soprano switched from the lyric to the dramatic role type.
topic / theme
'das Thema' and 'das Fach' can be closely related in school, university, or work contexts. 'Thema' means the specific content that you talk about or learn about. 'Fach' means a larger field of knowledge, for example mathematics, German, or biology. A subject contains many topics. That is why the words are easy to confuse: 'Geschichte' (history) is a 'Fach', but 'der Zweite Weltkrieg' (the Second World War) can be a 'Thema' in the subject of history. 'Fach' can also mean a specialty in a profession; 'Thema' remains the specific content of a conversation or lesson.

Biologie ist Sofias Lieblingsfach in der Schule.
Biology is Sofia's favorite subject at school.
Nach dem Unterricht war das neue Thema noch vielen Schülern unklar.
After the lesson, the new topic was still unclear to many students.