der Pudel
poodle
A breed of dog known for its characteristic curly coat and bred in different sizes.
Meine Nachbarin geht jeden Tag mit ihrem schwarzen Pudel spazieren.
My neighbor goes for a walk with her black poodle every day.

poodle
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poodle
A breed of dog known for its characteristic curly coat and bred in different sizes.
Meine Nachbarin geht jeden Tag mit ihrem schwarzen Pudel spazieren.
My neighbor goes for a walk with her black poodle every day.

When 'Pudel' (poodle) occurs in the fixed expressions 'pudelwohl' (feeling great) or 'des Pudels Kern' (the heart of the matter), it does not refer to the dog. These idiomatic expressions have a figurative meaning that becomes clear from context.
Zuerst schien das Problem einfach, aber dann zeigte sich des Pudels Kern.
At first the problem seemed simple, but then the heart of the matter emerged.
The expression 'sich pudelwohl fühlen' (to feel great) is reflexive. The person who feels good is the subject of the sentence, and 'pudelwohl' is an adverb that describes how someone feels.
Im neuen Haus mit dem großen Garten fühlt sich die ganze Familie pudelwohl.
In the new house with the large garden, the whole family feels great.
In 'pudelwohl' (feeling great), 'pudel-' serves as an intensifier and means 'very' or 'completely'. This intensifying function also appears in 'pudelnass' (soaking wet).
Nach dem Bad im See war der Hund pudelnass.
After swimming in the lake, the dog was soaking wet.
The idiom 'des Pudels Kern' (the heart of the matter) comes from Johann Wolfgang von Goethe's famous work 'Faust'. It is used to reveal the surprising, true essence of something.
Aha, das ist also des Pudels Kern! Du wolltest von Anfang an nur mein Geld.
Aha, so that's the heart of the matter! You only wanted my money from the start.
The expression 'des Pudels Kern' (the heart of the matter) is a fixed unit with a genitive ('des Pudels'), which is not changed. The whole phrase functions in the sentence like a normal noun.
Lange haben wir diskutiert, bis wir des Pudels Kern endlich verstanden.
We discussed it for a long time until we finally understood the heart of the matter.
dog
'der Hund' and 'der Pudel' both refer to an animal. 'Hund' is the general word for the animal species. 'Pudel' is a specific dog breed with typical curly fur. The words can be confused because every poodle is a dog. But not every dog is a poodle. If the breed does not matter or is unknown, you say 'Hund'. If you mean exactly this breed, 'Pudel' is correct.

Der Pudel meiner Nachbarin hat sehr lockiges, weißes Fell.
My neighbor's poodle has very curly white fur.
Im Park läuft ein Hund frei über die Wiese.
A dog is running freely across the grass in the park.