der Kater
tomcat
A male cat.
Unser Kater schläft am liebsten den ganzen Tag auf dem Sofa.
Our tomcat likes to sleep on the sofa all day.

tomcat / hangover
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tomcat
A male cat.
Unser Kater schläft am liebsten den ganzen Tag auf dem Sofa.
Our tomcat likes to sleep on the sofa all day.

You can easily recognize the meaning from context. If people are talking about animals, meowing, or food, it means the male cat. If it is about alcohol, a party the day before, or headaches, it means feeling unwell.
Unser Kater fängt gerne Mäuse im Garten.
Our tomcat likes to catch mice in the garden.
To express that you feel bad after too much alcohol, you use the fixed expression 'einen Kater haben' (to have a hangover). The verb 'haben' (to have) is central here.
Ich gehe heute nicht aus, ich habe noch einen Kater von gestern.
I'm not going out today; I still have a hangover from yesterday.
The meaning 'feeling unwell after drinking alcohol' is colloquial. In a formal or medical context, other words would be used.
Er rief bei der Arbeit an und sagte, er habe einen schlimmen Kater.
He called work and said he had a bad hangover.
To describe the intensity of feeling unwell, 'Kater' (hangover) is often combined with adjectives like 'schlimm' (bad), 'schrecklich' (terrible), 'ordentlich' (serious), or 'gewaltig' (massive).
Nach der Hochzeitsparty hatten viele Gäste am nächsten Tag einen gewaltigen Kater.
After the wedding party, many guests had a massive hangover the next day.
cat
'die Katze' and 'der Kater' belong to the same animal category. 'die Katze' can mean the animal in general, regardless of whether it is male or female. But it can also specifically mean a female animal if the sex is known. 'der Kater', on the other hand, always means a male cat. Learners may confuse the words because both are often used for pets in everyday life. If the sex is not important, people usually say 'Katze'. If you want to clearly say that the animal is male, you say 'Kater'.

Der Kater meiner Nachbarin ist sehr groß und hat schwarzes Fell.
My neighbor's tomcat is very large and has black fur.
Unsere Katze schläft am liebsten auf dem Sofa.
Our cat likes sleeping on the sofa best.