der Flieger
plane
(Colloquial) An airplane.
Unser Flieger nach Mallorca hat eine Stunde Verspätung.
Our plane to Mallorca is delayed by one hour.

plane / pilot
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plane
(Colloquial) An airplane.
Unser Flieger nach Mallorca hat eine Stunde Verspätung.
Our plane to Mallorca is delayed by one hour.

'Der Flieger' (plane/pilot) is derived from the verb 'fliegen' (to fly) with the ending '-er'. This ending can refer both to a person who does something (pilot) and to a device that does something (airplane).
Der Flieger ist sicher gelandet.
The plane landed safely.
You can recognize the meaning of 'Flieger' (plane/pilot) from the context. If it is about travel or takeoff and landing, it means an airplane. If it is about controlling the aircraft, it means a pilot. In a military context, it refers to air force soldiers.
Der Flieger nach Berlin startet in einer Stunde.
The plane to Berlin leaves in an hour.
When 'Flieger' means an airplane, it is a colloquial expression. In more formal contexts or in written language, 'das Flugzeug' (the airplane) is usually used.
Wann geht unser Flieger morgen früh?
When does our plane leave tomorrow morning?
In the meaning 'Flugzeug' (airplane), 'Flieger' (plane) is often combined with verbs of travel: 'den Flieger nehmen' (to take the plane), 'den Flieger verpassen' (to miss the plane), or 'im Flieger sitzen' (to be sitting on the plane).
Wir müssen uns beeilen, sonst verpassen wir den Flieger.
We have to hurry, otherwise we'll miss the plane.
machine
'die Maschine' and 'der Flieger' can both colloquially mean an airplane. Both words mainly fit in conversations about travel and flights. 'Flieger' sounds very casual and everyday. 'Maschine' is also used colloquially, but often sounds a bit more typical in sentences about takeoff, landing, or delays. Learners may confuse the words because neither is the neutral standard word. In addition, 'der Flieger' can also mean a person who flies an airplane. If you want to be clear and neutral, 'Flugzeug' is the clearest choice; between 'Maschine' and 'Flieger', the conversation context often decides.

Im Büro erzählt der Kunde, dass sein Flieger am Abend geht.
At the office, the customer says that his plane leaves in the evening.
Die Maschine aus Madrid ist pünktlich gelandet.
The plane from Madrid landed on time.