Pronouns are a handy way to keep from repeating nouns too much. Consider the following:

• When Big Dog snuck under the fence, Big Dog ran to the dumpster.
• When Big Dog snuck under the fence, he ran to the dumpster.

Clearly the second sentence sounds better.
When we talked about subject-verb agreement, we said that the subject and the verb must agree in number. That is, you have to make sure that both are singular or both are plural. The same goes for pronouns and antecedents.
A pronoun renames (takes the place of) a noun that comes before it.
An antecedent is what we call the noun that comes before the pronoun.
Two examples should be enough:

1. I get worried when the neighbors let their dog out.
2. The dog goes wild, and he always messes up my front yard.

In number 1, neighbors is the antecedent; their is the pronoun. They agree because both are plural. In number 2, dog is the antecedent, and he is the pronoun. They agree since both are singular. If you are having problems with pronoun-antecedent agreement, underline all your pronouns; then, locate the antecedent for each. Make sure that both are the same in number. That's all you have to do.
With subject-verb agreement, we also talked about indefinite pronouns ("-one," "-body," and "-thing" words). With pronouns, we need to look at the "-one" and "-body" words again. (Also include "either," "neither," and "each.") These are always singular, and that should be easy enough to remember. But most beginning writers seem to have trouble with them. Part of the problem is the way we speak. Few would notice anything wrong if they heard someone say,

• Everyone needs to take their dog to the vet regularly.
• Everybody was enjoying their vacation.

These sound okay, right? Maybe so, but they're wrong if you're writing. Words like "everyone" and "everybody" are always singular, so the plural pronoun their can't agree with them. The correct
versions are:

• Everyone needs to take his dog to the vet regularly.
• Everybody was enjoying her vacation.

You could use his or her in either sentence, depending upon the context.

Sexism in the language

Some suggest that you should always use "he or she," "his or her," or "him or her" instead of simply using the masculine "he," "his," or "him." If you're worried about the sexist tradition of using masculine pronouns to refer to all people, then rewrite your sentences in the plural, as in "People need to take their dogs to the vet regularly." "The people were enjoying their vacations."
You could use "he or she" (etc.), but it sounds awkward and will eventually lead to an agreement problem--if it doesn't bore your readers to death first. For instance:

If an owner doesn't take proper care of his or her dog, then he or she will find his or her pet may suffer. The pet will be grateful to him or her if he or she takes care of his or her reponsibilities. When he or she . . . .

You may be an inexperienced writer, but you're not an inexperienced listener. That passage simply sounds dumb. Like I said, if you're worried about a sexist slant, go for the plurals!