einen Kater haben

You use it just like any other physical condition. Use the verb haben (to have) followed by einen Kater. If the hangover is particularly brutal, you can add adjectives. A mordsmäßiger Kater is a 'murderous' one. You can also use it as a verb: katzen (to feel hungover), though this is much less common than the noun phrase. Just remember to conjugate haben correctly for whoever is suffering.

Use this with friends, family, and close colleagues. It is perfect for Sunday morning brunch conversations. You can use it when declining an invitation to go for a run. It works well in texts when you are explaining why you are still in bed at noon. It is a relatable, human expression that almost every adult German understands.

Avoid using this in very formal professional settings. Do not tell your CEO you have a Kater during a Monday morning board meeting. In those cases, say you are 'unpässlich' (unwell) or have a 'Migräne' (migraine). Also, do not use it for general sickness. If you have the flu, you don't have a Kater. You only have a Kater if alcohol was the primary cause of your misery.

The word actually comes from a misunderstanding! In the 19th century, students used the word Katarrh (catarrh), which is a medical term for inflammation. Because they pronounced it with a Saxon accent, it sounded like Kater (male cat). The name stuck because it was funnier than a medical term. Now, the 'hangover cat' is a staple of German drinking culture.

You might hear people talk about a Filmriss. That is a 'film tear' or a blackout from drinking. Another common one is Konterbier. This is the 'counter-beer' you drink the next morning to fight the Kater. If someone is acting very grumpy or hungover, you might say they are verkatert. This is the adjective form of the noun.