die Besatzung
crew
The entire team of a vehicle, especially a ship, airplane, or spacecraft.
Die Besatzung des Flugzeugs bestand aus zwei Piloten und vier Flugbegleitern.
The crew of the airplane consisted of two pilots and four flight attendants.

crew / occupation / trim
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crew
The entire team of a vehicle, especially a ship, airplane, or spacecraft.
Die Besatzung des Flugzeugs bestand aus zwei Piloten und vier Flugbegleitern.
The crew of the airplane consisted of two pilots and four flight attendants.

'Besatzung' (crew/occupation/trim) comes from the verb 'besetzen' (to occupy, to staff, to trim). The different meanings of the verb—to take up a place, to control a territory, to decorate a garment—lead directly to the meanings of the noun: crew, military occupation, and clothing trim.
Die Demonstranten besetzten den Platz vor dem Rathaus.
The demonstrators occupied the square in front of the town hall.
The three main meanings are very different. The context always makes clear what is meant: vehicles have a 'Besatzung' (crew), countries have a 'Besatzung' (occupation), and clothing has a 'Besatzung' (trim).
Die gesamte Besatzung konnte das sinkende Schiff rechtzeitig verlassen.
The entire crew was able to leave the sinking ship in time.
In the meaning of 'crew', 'Besatzung' is almost always used in connection with vehicles. Typical terms are 'Flugzeugbesatzung' (aircraft crew), 'Schiffsbesatzung' (ship's crew), or the 'Besatzung eines Raumschiffs' (crew of a spacecraft).
Die Besatzung bereitete das Flugzeug für den Start vor.
The crew prepared the airplane for takeoff.
In a military or political context, 'Besatzung' (occupation) is often used with words like 'militärisch' (military) or in compounds like 'Besatzungsmacht' (occupying power) or 'Besatzungszone' (occupation zone).
Er forscht zur Geschichte der französischen Besatzungszone in Deutschland.
He researches the history of the French occupation zone in Germany.
team / crew
'die Mannschaft' and 'die Besatzung' can mean the same thing for ships, airplanes, or spacecraft: the people who work there and look after or operate the vehicle. 'Besatzung' is very precise and typical in this context. 'Mannschaft' is a bit more general and can also be used for other shared tasks. Learners may confuse the words because they are often interchangeable for vehicles. But in sports, you say 'Mannschaft', not 'Besatzung'.

Die Besatzung des Flugzeugs begrüßt die Passagiere an Bord.
The crew of the airplane welcomes the passengers on board.
Die Mannschaft des Rettungsboots half den Menschen sofort.
The rescue boat crew helped the people immediately.