hauen
to hit
Colloquial for to strike; to give someone or something a forceful blow, often with the hand or an object.
Wenn du mich weiter ärgerst, hau ich dir eine!
If you keep annoying me, I'll hit you!

to hit / to carve
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to hit
Colloquial for to strike; to give someone or something a forceful blow, often with the hand or an object.
Wenn du mich weiter ärgerst, hau ich dir eine!
If you keep annoying me, I'll hit you!

The verb 'hauen' is mostly used colloquially for 'schlagen' (to hit/strike). One exception is the meaning 'formen' (to shape) or 'meißeln' (to chisel), which is common in the specialized language of crafts, e.g. among sculptors.
Pass auf, was du sagst, sonst hau ich dich!
Watch what you say, or I'll hit you!
When you hit a person, that person is often in the dative. A very common threat is the expression 'jemandem eine hauen' (to give someone one), where 'eine' stands for 'einen Schlag' (a blow/punch).
Er hat seinem Bruder aus Wut eine gehauen.
He hit his brother out of anger.
The reflexive form 'sich hauen' means that two or more people hit each other, that is, fight with each other. It is often used to describe a brawl or scuffle.
Die Kinder haben sich im Sandkasten gehauen.
The children fought in the sandbox.
The expression 'jemanden übers Ohr hauen' (to rip someone off) is a fixed idiom and means to cheat someone. The person who is cheated is always in the accusative.
Ich glaube, der Händler hat mich mit diesem Preis übers Ohr gehauen.
I think the dealer ripped me off with this price.
'Auf den Putz hauen' (to go all out) is a colloquial idiom. It means that someone parties exuberantly, shows off, or spends a lot of money.
Nach den Prüfungen haben wir am Wochenende ordentlich auf den Putz gehauen.
After the exams, we really went all out over the weekend.
to fight
'kämpfen' and 'hauen' can both have to do with physical violence. 'kämpfen' tends to describe the whole confrontation between opponents. This can be serious, athletic, or military. 'hauen' is colloquial and mainly means to hit. In the form 'sich hauen', it means that people are fighting or brawling. The words can be confused when talking about a physical confrontation. The important point is: 'kämpfen' describes the conflict as a whole; 'hauen' describes more the blow itself or a rough, colloquial fight.

Nach dem Streit haben sich zwei Jugendliche auf dem Schulhof gehauen.
After the argument, two teenagers got into a fight in the schoolyard.
Im Finale kämpfen die beiden Sportler um den Sieg.
In the final, the two athletes are fighting for victory.