Learn Spanish (Spain): Ayer tuve una pelea con mi hermano. - Yesterday I had a fight with my brother.

Because the fight was a specific, completed event that happened yesterday. You use the preterite tuve for completed actions, whereas you would use the imperfect tenía for ongoing situations in the past, like background descriptions.

Does Spanish always use tener for having a fight?

Yes, just like in English, Spanish uses tener (to have) to express having a fight or argument. The phrase is exactly tener una pelea.

Is pelea related to a verb?

Yes, it comes from the verb pelear, which means to fight. It is very common in Spanish for nouns to be formed directly from verbs by changing the ending.

What is the difference between una pelea and una discusión in this context?

While both can translate to a fight or argument, una discusión is strictly verbal. Una pelea is usually more aggressive and can mean a heated shouting match or even a physical fight.