
der Aufenthalt
stay
Staying or being in a particular place for a certain period of time.
Unser Aufenthalt in den Bergen war sehr erholsam.
Our stay in the mountains was very relaxing.
stay / residence
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stay
Staying or being in a particular place for a certain period of time.
Unser Aufenthalt in den Bergen war sehr erholsam.
Our stay in the mountains was very relaxing.

residence
The legal permission for a person without a country's citizenship to reside there. Often understood as part of 'Aufenthaltserlaubnis' (residence permit) or 'Aufenthaltsgenehmigung' (residence permit).
Für die Einreise und einen längeren Aufenthalt benötigen Sie ein Visum.
For entry and a longer stay, you need a visa.
The noun 'der Aufenthalt' (stay/residence) comes from the reflexive verb 'sich aufhalten' (to stay, to be located somewhere). If you stay in a place, then you have an 'Aufenthalt' there.
Wo hältst du dich am liebsten auf, wenn du frei hast?
Where do you like to spend time when you have time off?
In the meaning of 'Verweilen' (staying), 'Aufenthalt' (stay) is often combined with adjectives like 'kurz' (short), 'lang' (long), or 'angenehm' (pleasant). The typical preposition for the place is 'in', and for people it is 'bei'.
Wir wünschen Ihnen einen angenehmen Aufenthalt in unserem Hotel.
We wish you a pleasant stay in our hotel.
In connection with laws and authorities (especially the Ausländerbehörde, the foreigners' registration office), 'Aufenthalt' (residence) means the right to live in a country. It is often part of compounds such as 'Aufenthaltserlaubnis' (residence permit) or 'Aufenthaltsstatus' (residence status).
Er hat eine unbefristete Aufenthaltserlaubnis für Deutschland beantragt.
He applied for a permanent residence permit for Germany.
visit / visitor
Besuch and Aufenthalt can both refer to spending a limited amount of time in a place. Besuch emphasizes going to the destination, while Aufenthalt emphasizes staying there. Use Besuch when the journey to or purpose of the visit is important, and Aufenthalt when the time spent there is the focus.

Der Aufenthalt im Museum dauerte wegen der neuen Ausstellung zwei Stunden.
Our stay at the museum lasted two hours because of the new exhibition.
Der Besuch im Museum führte uns zu einer neuen Ausstellung.
The visit to the museum led us to a new exhibition.