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.

'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 do / make
'machen' and 'betreiben' can both express that someone carries out an activity. However, 'machen' is very general and fits many everyday situations. 'betreiben' sounds more specific and often somewhat more formal. It is used when someone does something regularly, for example a sport, or when someone actively keeps a business, website, or project running. As a simple guide: in everyday life, people often say 'machen'; for regular, organized, or serious activity, 'betreiben' is often a better fit.

Meine Nachbarin betreibt seit zwei Jahren einen kleinen Online-Shop.
My neighbor has been running a small online shop for two years.
Nach der Arbeit mache ich noch schnell die Hausaufgaben mit meinem Sohn.
After work, I quickly do homework with my son.