die Neigung
inclination
A personal preference, bent, or tendency toward a particular activity, behavior, or feeling.
Er hat eine starke Neigung zur Musik.
He has a strong inclination toward music.

inclination
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inclination
A personal preference, bent, or tendency toward a particular activity, behavior, or feeling.
Er hat eine starke Neigung zur Musik.
He has a strong inclination toward music.

The noun 'Neigung' (inclination) is derived from the verb 'neigen' (to incline / to tend). The verb means both 'to hold something at an angle' and 'to tend toward something'. This double meaning is also found in the noun.
Der alte Baum neigt sich gefährlich über die Straße.
The old tree is leaning dangerously over the road.
The meaning of 'Neigung' (inclination) becomes clear from the context. If it is about physics or construction, it refers to a slope. If it is about a person, it refers to affection. With behavior or hobbies, it means a tendency.
Die Neigung der Rampe ist für Rollstuhlfahrer zu steil.
The slope of the ramp is too steep for wheelchair users.
To express a personal tendency, you use the fixed structure 'eine Neigung zu etwas haben' (to have an inclination toward something). 'Zu' is always followed by the dative.
Er hat eine Neigung zu schnellen Autos.
He has an inclination toward fast cars.
When 'Neigung' means affection for a person, it is often combined with the verbs 'fühlen' (to feel) or 'empfinden' (to feel / to experience).
Er hat schon lange eine heimliche Neigung für sie empfunden.
He had long felt a secret affection for her.
desire / pleasure
'die Lust' and 'die Neigung' both have to do with an inner desire or preference. 'Lust' usually refers to a current feeling: you want to do something now or soon because it sounds pleasant. 'Neigung' describes more of a fixed personal tendency or preference that lasts longer. That is why 'Lust' sounds more natural in everyday sentences like 'Ich habe Lust auf Kino' (I feel like going to the movies). 'Neigung' fits better when talking about character, interests, or typical behavior.

Ich habe eine Neigung dazu, wichtige Aufgaben zu früh zu planen.
I have a tendency to plan important tasks too early.
Sie hat heute keine Lust zu kochen und bestellt etwas.
She doesn't feel like cooking today and orders something.