Learn Spanish (Latin America): La pareja se peleaba cuando llegué.

Even though la pareja refers to two people, it is a grammatically singular noun in Spanish. Because of this, it takes the singular pronoun se and the singular verb form peleaba.

Is se the word that means 'each other' here?

Exactly! The pronoun se shows that the action is reciprocal between the two people in the couple. Without it, peleaba would just mean someone was fighting in general, but se peleaba means they were fighting each other.

Why do we use two different past tenses (peleaba and llegué) in the same sentence?

This is a classic past-tense combination! We use the imperfect tense for the background action that was already going on (peleaba, meaning 'was fighting'). We use the preterite tense for the sudden event that interrupted it (llegué, meaning 'arrived').

I know the verb is llegar. Why is it spelled with a 'u' in llegué?

In Spanish, a 'g' followed by an 'e' makes a soft 'h' sound. To keep the hard 'g' sound from llegar, we have to add a silent 'u' before the 'e'. This spelling change happens with all verbs ending in '-gar' when using the 'I' form in the preterite past tense.