Learn Italian: Il professore ci ha spiegato i teoremi. - The professor explained the theorems to us.

Teorema is one of those tricky masculine nouns of Greek origin that end in -ma in the singular (il teorema). Because it is a masculine noun, its plural form ends in -i, making it i teoremi, and it takes the masculine plural article i.

Is ci a direct or indirect object in this sentence?

It is an indirect object. The verb spiegare (to explain) takes a direct object for the thing being explained (i teoremi) and an indirect object for the person receiving the explanation. Here, ci stands for to us.

Why does ci come before the verb ha spiegato?

In Italian, object pronouns like ci usually go directly before the conjugated verb. Because this sentence is in the passato prossimo, ci is placed immediately before the auxiliary verb ha.

Shouldn't the past participle spiegato change to spiegati to match i teoremi?

No, because the sentence uses the auxiliary verb avere (ha) and the direct object (i teoremi) comes after the verb. The past participle only agrees with the direct object when a direct object pronoun (such as li) is used before the verb.