
die Firma
company
A company or business enterprise that manufactures products or offers services.
Sie hat eine eigene Firma für Webdesign gegründet.
She founded her own company for web design.
company / firm
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company
A company or business enterprise that manufactures products or offers services.
Sie hat eine eigene Firma für Webdesign gegründet.
She founded her own company for web design.

business name
The official name of a merchant or company entered in the commercial register, under which the business is conducted.
Die Firma des Unternehmens lautet „Schulze & Partner KG“.
The business name of the company is 'Schulze & Partner KG'.
When talking about an employer, you can use both 'in' and 'bei'. 'Ich arbeite in einer großen Firma' (I work in a large company) emphasizes the organization, while 'Ich arbeite bei Siemens' (I work at Siemens) highlights the name of the employer.
Er arbeitet seit zehn Jahren in derselben Firma.
He has worked in the same company for ten years.
Common verbs used with 'Firma' (company) in the meaning of 'Unternehmen' (enterprise/company) are 'eine Firma gründen' (to found a company), 'eine Firma leiten' (to run a company), or 'eine Firma übernehmen' (to take over a company).
Nach dem Studium wollte er sofort eine eigene Firma gründen.
After finishing his studies, he wanted to found his own company right away.
The meaning 'official name' is technical language from the fields of business and law. You can often recognize this meaning from verbs such as 'lauten' (to read / to be worded) or in connection with the 'Handelsregister' (commercial register).
Die Firma des Unternehmens lautet „Müller & Sohn GmbH“.
The business name of the company is 'Müller & Sohn GmbH'.
company / undertaking
In everyday usage, das Unternehmen and die Firma can refer to the same business organization. However, Firma can also specifically mean its officially registered name. Use Unternehmen clearly for the organization; Firma is more precise when referring to the official name.

Die Firma des Unternehmens lautet offiziell Meditec GmbH.
The company’s official name is Meditec GmbH.
Das Unternehmen entwickelt Geräte für Krankenhäuser.
The company develops equipment for hospitals.