
die Beschäftigung
employment
Paid work or a professional activity that someone does.
Er hat nach langer Suche endlich eine neue Beschäftigung gefunden.
After a long search, he finally found new employment.
employment / activity
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employment
Paid work or a professional activity that someone does.
Er hat nach langer Suche endlich eine neue Beschäftigung gefunden.
After a long search, he finally found new employment.

activity
An activity with which one spends one's time, often for pleasure or as a hobby.
Lesen ist meine liebste Beschäftigung am Abend.
Reading is my favorite activity in the evening.

engagement with
The intensive engagement with a particular topic or question.
Die Beschäftigung mit der deutschen Geschichte ist ein wichtiger Teil des Lehrplans.
Engagement with German history is an important part of the curriculum.
'Beschäftigung' (employment/activity) is a noun derived from the verb 'sich beschäftigen' (to occupy oneself / to engage with something). The ending '-ung' turns the activity into a noun that describes the activity itself or its result.
Womit beschäftigst du dich in deiner Freizeit?
What do you occupy yourself with in your free time?
In the meaning of 'Arbeit' (work) or 'Job' (job), 'Beschäftigung' (employment) is mainly used in formal, official, or economic contexts, for example in employment contracts or statistics.
Er sucht eine geringfügige Beschäftigung für das Wochenende.
He is looking for marginal employment for the weekend.
When referring to a leisure activity, 'Beschäftigung' (activity) is often combined with adjectives like 'sinnvoll' (meaningful), 'kreativ' (creative), or 'entspannend' (relaxing). It can also be referred to as a 'Lieblingsbeschäftigung' (favorite activity).
Gartenarbeit ist für mich eine entspannende Beschäftigung.
Gardening is a relaxing activity for me.
When it is about engaging with a topic, the preposition 'mit' (with) and the dative are almost always used. The pattern is: 'die Beschäftigung mit [Thema]' (engagement with [topic]).
Die Beschäftigung mit der eigenen Vergangenheit kann sehr aufschlussreich sein.
Engagement with one's own past can be very revealing.
work
Arbeit and Beschäftigung can both refer to paid professional work. Arbeit is the everyday term for a job or work someone has done; Beschäftigung sounds more formal and focuses more on the employment someone is engaged in. Beschäftigung is therefore often used on forms or when referring to employment status.

Im Antrag weist Lea ihre Beschäftigung in einer Bäckerei nach.
In the application, Lea provides proof of her employment at a bakery.
Lea erzählt ihren Freunden, dass sie endlich Arbeit in einer Bäckerei gefunden hat.
Lea tells her friends that she has finally found a job at a bakery.