Agreement means consistency. That is, if a noun is singular, it takes a singular verb form or pronoun; if a noun is plural, it takes a plural verb form or pronoun.
Here are some examples of noun-verb agreement.
I am at home. ("am" agrees with the subject, "I")
She is at home. ( "is" agrees with the subject, "she")
The girls are at home. ("are" agrees with the subject, "the girls")
Pronouns take the place of nouns. If a pronoun replaces a subject, use these pronouns: I, she, he, it, they, we.
My brother and I will go to Florida in May.
If pronouns come after a preposition in a sentence, use these pronouns: me, her, him, it, them, us.
Send a reply to Sally and me.
The book belongs to him.
Here is a typical pronoun error:
Me and her went to the movies.
"Me" and "her" cannot be used as the subject of a sentence. Therefore, the correct sentence is
She and I went to the movies.
Here's another example of the same type of error:
The pizza was divided between my roommate and I.
Since the pronoun comes after the preposition "between," the sentence should read
The pizza was divided between my roommate and me.
A singular pronoun should replace a singular noun; a plural pronoun should replace a plural noun. Here, for example, is a very common pronoun mistake with a few possible revisions:
Any Skidmore student can decorate their dorm room with posters.
This is grammatically incorrect because "Any Skidmore student" refers to a single student, and the pronoun "their" is a plural possessive pronoun. (In the same way, "each Skidmore student" refers to a single student, not to everyone.) There are two ways to revise such a sentence:
Any Skidmore student can decorate his or her dorm room with posters.
Skidmore students can decorate their rooms with posters.
If you want to avoid gender bias in choosing pronouns, changing a noun to a plural can help solve the problem. In general, overusing "his or her" construction makes writing awkward.
Click here to read more about agreement in the context of correct grammar.