Learn French: Nous nous sommes disputés et il a crié.

The first nous is the subject ("we"). The second nous is a reflexive pronoun meaning "each other." The verb being used is se disputer, which means "to argue with each other."

Why is the second "nous" placed before "sommes" instead of right before "disputés"?

In the passé composé, pronoun objects always go right in front of the conjugated helping verb (the auxiliary). Just like the indirect object pronouns you are currently learning, the reflexive nous must go before sommes, never before the past participle (disputés).

Why does "disputés" have an "s" at the end, while "crié" does not?

Reflexive verbs like se disputer always use être as their helping verb in the past tense. With être, the past participle usually agrees with the subject, so we add an s because nous is plural. Crier (to shout) uses avoir, which doesn't add an s to match the subject.

Does "se disputer" just mean "to discuss"?

No, this is a common false friend for English speakers! Se disputer specifically means to argue, fight, or quarrel. If you want to say "to discuss" in a calm or normal way, you would use the verb discuter.