
betreiben
to operate
To run and keep a business, facility, website, or organization going.
Meine Tante betreibt ein kleines Hotel an der Küste.
My aunt operates a small hotel on the coast.
to operate / pursue
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to operate
To run and keep a business, facility, website, or organization going.
Meine Tante betreibt ein kleines Hotel an der Küste.
My aunt operates a small hotel on the coast.

to pursue
To regularly and actively engage in an activity, hobby, or sport.
Er betreibt in seiner Freizeit intensiv Ahnenforschung.
He pursues genealogy intensively in his free time.

to push for
To pursue or push something forward with determination and effort in order to achieve a specific goal.
Die Anwohner betreiben die Einrichtung einer Tempo-30-Zone in ihrer Straße.
The residents are pushing for the creation of a 30 km/h zone on their street.
'Betreiben' (to operate / pursue) is an inseparable verb with the prefix 'be-'. The base verb is 'treiben' (to drive something forward). So 'betreiben' means to actively 'drive forward' an activity or business and keep it going.
Sie betreibt den Blog schon seit fünf Jahren mit viel Leidenschaft.
She has been running the blog with a lot of passion for five years.
'Betreiben' (to operate / pursue) is a transitive verb and almost always requires an accusative object. The question is always: whom or what does someone operate/pursue?
Welche Art von Geschäft betreiben Sie?
What kind of business do you operate?
You can recognize the exact meaning of 'betreiben' (to operate / pursue) from the object: Is it a business ('ein Café betreiben' — to run a café), a hobby ('Sport betreiben' — to do sports), or an abstract goal ('Aufwand betreiben' — to make an effort)?
Er betreibt ein kleines Kino in der Innenstadt.
He operates a small movie theater in the city center.
In the business sense, 'betreiben' (to operate) is often combined with places or organizations: 'ein Restaurant betreiben' (to run a restaurant), 'ein Hotel betreiben' (to run a hotel), 'eine Webseite betreiben' (to operate a website), or 'eine Firma betreiben' (to run a company).
Die Familie betreibt seit Generationen einen erfolgreichen Bauernhof.
The family has operated a successful farm for generations.
When talking about leisure activities or science, 'betreiben' (to pursue) is used for serious and regular activities. Typical phrases are: 'Sport betreiben' (to do sports), 'ein Hobby betreiben' (to pursue a hobby), or 'Forschung betreiben' (to conduct research).
Welchen Sport betreibst du in deiner Freizeit?
What sport do you do in your free time?
to lead
Both words work when someone is responsible for a business. „Führen“ particularly emphasizes the management role and organizing people, while „betreiben“ highlights running the business on an ongoing basis and keeping it operating. „Führen“ is usually more suitable for being responsible for a department, while „betreiben“ is more suitable for the practical operation of a business.

Frau Keller betreibt das Restaurant und hält es jeden Tag geöffnet.
Ms. Keller operates the restaurant and keeps it open every day.
Frau Keller führt das Restaurant und organisiert das gesamte Team.
Ms. Keller runs the restaurant and organizes the entire team.