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.

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 describe time spent at a place. 'Besuch' emphasizes that you go to a person or a place, often for a specific purpose or for social reasons. 'Aufenthalt' only emphasizes that you stay in a place for a certain amount of time. It sounds more natural with hotels, hospitals, travel, or official situations. You can visit someone and have a short stay at the same time. If the meeting or the act of going there is important, 'Besuch' fits. If the duration or the place is important, 'Aufenthalt' fits better.

Während seines Aufenthalts im Krankenhaus bekam Paul jeden Tag Besuch.
During his stay in the hospital, Paul had visitors every day.
Der Besuch bei meiner Nachbarin war kurz, aber sehr nett.
The visit to my neighbor was short but very nice.