field
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field / specialty
A specialized area within a science, profession, or area of knowledge in which a person has special expertise.
Ihr Fachgebiet ist die Astrophysik, insbesondere die Erforschung schwarzer Löcher.
Her specialty is astrophysics, especially the study of black holes.
The word is made up of 'das Fach' (here: specialization, discipline) and 'das Gebiet' (area, field). So it literally means a 'field of specialization.'
Sein Fach ist Geschichte, aber sein spezielles Gebiet ist das Römische Reich.
His subject is history, but his specific field is the Roman Empire.
The phrase 'Das ist (nicht) mein Fachgebiet' (That is/is not my field) is used very often to express that you are or are not familiar with a topic. The specific area is often attached with the genitive.
Fragen Sie dazu bitte meine Kollegin, das ist nicht mein Fachgebiet.
Please ask my colleague about that; it's not my field.
Typical verbs in connection with 'Fachgebiet' (field/specialty) are 'sich auf ein Fachgebiet spezialisieren' (to specialize in a field), 'in einem Fachgebiet arbeiten' (to work in a field), or 'ein Fachgebiet beherrschen' (to master a field).
Nach dem Studium hat sie sich auf das Fachgebiet des Arbeitsrechts spezialisiert.
After her studies, she specialized in the field of labor law.
area / field
'der Bereich' and 'das Fachgebiet' can both mean a topic or field of knowledge. 'Bereich' is very general and fits in many situations, for example in work, business, or science. 'Fachgebiet' sounds more specific and means a specialized field in which someone has particular expertise. The words can be confused because both talk about topics. When it is about expert knowledge or a subject in studies or a profession, 'Fachgebiet' is often more suitable. If the topic is only being named broadly, 'Bereich' is enough.
Das Fachgebiet von Frau Schwarz ist die Kinderpsychologie.
Ms. Schwarz's specialty is child psychology.
Er arbeitet im Bereich Umwelt und Energie.
He works in the field of environment and energy.