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.

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'.
house
'das Haus' and 'die Firma' can both mean a company. 'Firma' is the normal word when referring to a business that offers goods or services. In this meaning, 'Haus' often sounds more elevated or traditional and fits especially well in industries such as fashion, publishing, or finance. In everyday language, people usually say 'Firma'. 'Haus' is used more when you want to present the company as a well-known brand or institution.

Meine Firma bietet jetzt auch Beratung per Video an.
My company now also offers consultations by video.
Das Haus bringt im Herbst eine neue Modekollektion heraus.
The house is releasing a new fashion collection in the fall.