Katze vs. Kater
In German, both Katze and Kater translate to cat, but they are not interchangeable. The primary difference lies in their reference to the cat's gender and their use in a general context. Katze is the general term, while Kater is gender-specific.
Katze
A1The word Katze (a feminine noun, die Katze) is the general, all-purpose term for a cat. You use it when the cat's gender is unknown, irrelevant, or when referring to cats as a species. It is also the specific term for a female cat.
Ich sehe eine Katze im Garten.
(I see a cat in the garden.)
Die Katze ist ein sehr unabhängiges Tier.
(The cat is a very independent animal.)
Unsere Katze hat letztes Jahr Junge bekommen.
(Our cat had kittens last year.)
Als Kind wollte ich immer eine Katze haben.
(As a child, I always wanted to have a cat.)
Der Tierarzt untersucht die kleine Katze.
(The veterinarian is examining the little cat.)
Kater
A2The word Kater (a masculine noun, der Kater) specifically refers to a male cat, also known as a tomcat. It is only used when you know the cat is male. Additionally, Kater is a common colloquial term for a hangover.
Unser Kater schläft den ganzen Tag.
(Our male cat sleeps all day.)
Der schwarze Kater der Nachbarn ist sehr groß.
(The neighbors' black male cat is very large.)
Ist das eine Katze oder ein Kater?
(Is that a female cat or a male cat?)
Die beiden Kater kämpfen oft miteinander.
(The two male cats often fight with each other.)
Nach der Feier gestern habe ich heute einen Kater.
(After the party yesterday, I have a hangover today.)
Summary
In essence, use Katze as the default, general word for any cat, much like cat in English. It is also the correct term for a female cat. Use Kater exclusively when you are referring to a male cat. Therefore, every Kater is also a Katze in the general sense, but not every Katze is a Kater. Finally, remember the secondary, informal meaning of Kater as a hangover.