Agreements in sentences refer to the subjects agreeing with their verbs, and pronouns agreeing with their antecedents.
Subjects and their verbs
The subject and the main verb within a sentence must agree with each another in number, which is to say if a subject is singular, the verb must also be singular. Likewise, if a subject is plural, the verb must also be plural.
The subject and the main verb within a sentence must agree with each another in person as well. There are first person, second person, and third person.
Pronouns and their antecedents
A pronoun replaces an earlier noun which is called antecedent. The pronoun must agree with the antecedent it replaces in number (singular or plural), person (first, second, or third), and gender (masculine, feminine, or neuter).
Number: The pronoun must agree with the antecedent it replaces in number. This means a singular pronoun must replace a singular noun; and a plural pronoun must replace a plural noun.
A dog savaged a five-year-old child. Police have confirmed it was later destroyed.
My brother and I had a real fight this morning. We later apologized to each other.
A stranger approached me for a cigarette. He looked like my cellmate in prison.
The father and son are barbers in the same shop. They are in fact owners of the shop.
Gender: A singular feminine pronoun must replace a singular feminine noun, and a singular masculine pronoun must replace a singular masculine noun.
The princess is getting engaged next month. Rumour has it that she will be engaged to a commoner.
Abu has three camels. He inherited them from his father.