Infinitives are verbal abstract nouns

They are not limited by person or number; hence, their name: without limit (finis) or definition. They are considered neuter in gender but they cannot be declined. They are found only in the nominative and the accusative case (and like all nouns, pronouns and adjectives they have the same form in the nominative and the accusative). They are, of course, singular. They can be used as the subject of a verb: errare est humanum = "to err is human." (Note humanum = neuter singular, in agreement with the subject, errare). Infinitives are also used as the object of a verb. They are commonly found as the object of a verb of wish or desire: volo ridere = "I want to laugh." Compare: Quid vis? Volo domum ire. = "What do you want? I want to go home." quid is the direct object of vis, just as domum ire is the direct object of volo. Some expressions like this are sometimes called Complementary Infinitives, if it is felt that the meaning of the introductory verb must be completed by an infinitive. Other verbs and verbal expressions take the infinitive as a direct object as well: animum induco = I put it in my mind, or I intend; the action you intend can then made the direct object of the verbal expression. Similarly, conari desisti = "I stopped trying".

Now, if I wanted to say "I want him to go home," I would have to specify the subject of the infinitive, and in Latin I must do that in the accusative case. It is called the Subject Accusative of the Infinitive: volo eum domum ire; and its development is outlined on the  subject accusative page. When the infinitive is used as the direct object of a verb of speaking, thinking, perceiving, etc., the construction is called indirect discourse and the most important rule is the the subject of an infinitive in indirect discourse must always be stated (in the subject accusative construction). dixi eum domum ire = "I said that he was going home." You may want to review Tenses and the Infinitive in Indirect Discourse.