die Flamme
flame
The visible, glowing, gaseous part of a fire.
Die Flamme der Kerze tanzte im leichten Windzug.
The candle's flame danced in the light draft.

flame
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flame
The visible, glowing, gaseous part of a fire.
Die Flamme der Kerze tanzte im leichten Windzug.
The candle's flame danced in the light draft.

The noun 'die Flamme' (flame) belongs to a word family with the verb 'flammen' (to glow, to burn) and the adjective 'flammend' (flaming, e.g. flaming red). Recognizing this family helps with understanding related words.
Das Feuer im Kamin flammte gemütlich vor sich hin.
The fire in the fireplace flickered cozily.
In connection with fire, 'Flamme' (flame) is often used in fixed expressions. Typical ones are 'in Flammen stehen' (to be in flames/burning) or 'eine Flamme entzünden' (to ignite a flame/start a fire).
Das verlassene Gebäude stand plötzlich in Flammen.
The abandoned building was suddenly in flames.
When 'Flamme' (flame) describes a person, this is colloquial. People often speak of 'einer alten Flamme' (an old flame/former love interest) or 'einer neuen Flamme' (a new flame/current love interest).
Er hat seine alte Flamme aus der Jugend zufällig im Supermarkt getroffen.
He happened to meet his old flame from his youth in the supermarket.
To describe a strong feeling, 'Flamme' (flame) is often combined with a genitive that names the feeling. This use is rather elevated or poetic. Typical examples are 'die Flamme der Leidenschaft' (the flame of passion) or 'die Flamme des Zorns' (the flame of anger).
In ihren Augen loderte die Flamme der Entschlossenheit.
The flame of determination blazed in her eyes.
fire
'das Feuer' and 'die Flamme' both belong to the area of burning, heat, and light. 'Feuer' refers to the whole event or phenomenon: something is burning, there is heat, and often several flames. 'Flamme' refers more specifically to the single visible, glowing part of a fire. That is why you can speak of a 'Flamme' with a candle, but with a campfire you are more likely to speak of 'Feuer'. The words can be confused because you see flames when a fire is burning. There is also a connection with feelings: 'Feuer und Flamme sein' means to be very enthusiastic, but 'Flamme' by itself is not always simply the same as 'Feuer'.

Die Flamme der Kerze wurde im offenen Fenster immer kleiner.
The candle's flame grew smaller and smaller in the open window.
Im Kamin brannte ein kleines Feuer, als wir nach Hause kamen.
A small fire was burning in the fireplace when we came home.