The accusative case is used for the direct object of transitive verbs, for the internal object (mostly of intransitive verbs), for the subject of a subordinate infinitive (that is, not as the subject of the historical infinitive), to indicate place to which, extent or duration, and for the object of certain prepositions. In the masculine and feminine singular it always ends in -m; (cp. English: whom, him); in the masculine and feminine plural, it always ends in -s; and in the neuter plural, it always ends in -a. In English we do not have an accusative case as such; rather, we have the accusative function of the Object Case.
It is believed that the accusative case originally had a "local" function; it was the case that indicated the end or ultimate goal of an action or movement. Take an example: "I'm gonna hit your face." Here, "your face" is the end or the ultimate goal of my hitting and so it goes into the accusative case. This is the origin of the Direct Object. Another example from the classical world: the Latin peto originally meant "I fly" and referred to swift, eager movement. We know this in part because the Greek word, petomai, is related and it means "I fly." If I "move swiftly and eagerly" and the end of my motion is "Rome", then I can say peto Romam. Such an expression developed in Latin to mean "I seek Rome" and a verb originally intransitive ("I fly, move swiftly") became transitive ("I seek."). Compare now: Romam eo. Here, eo is still an intransitive verb meaning "I go" and Romam is still the "end of motion" (not the direct object).
The accusative of place to which is a vestige of the original meaning of the accusative case. Usually, the "place to which" is made the object of a preposition, but in the cases of cities, towns and small islands, of domus and of rus the accusative case is used alone: Veronam venis? = "Are you coming to Verona?" How small does the small island have to be? We have a similar, but more limited rule, in English. Do you know what it is?
The basic function of specifying the end of movement means that the Accusative is attached especially to verbs of motion and to prepositions when they refer to motion.
There are several metaphorical extensions of the basic function of the accusative, the most important being its use either alone or with the preposition ad to express the purpose of an action. Without a preposition, one finds the supine in the accusative case used after verbs of motion to express purpose. With the gerund and the gerundive, the preposition ad is used with the accusative to express purpose. And in many prepositional phrases with the accusative, the idea of purpose is more prominent than the idea of motion: ad salutem = "for safety, for the purpose of creating safety" etc.
Internal Accusative: Another important usage for the accusative case is the Internal Accusative. The Internal Accusative is any accusative that names or modifies the action of the verb. The Cognate Accusative is the easiest form of the internal accusative to identify; it is called a "cognate accusative" because the noun in the accusative case uses a same linguistic stem or root as (in other words, it is cognate with) the stem or root of the verb. I sing a song = intransitive verb, "I sing", + accusative that simply renames the activity of singing, "a song." This cognate (internal) accusative can be modified by adjectives: I sing a loud song. But notice what happens if we leave off the noun: I sing loud. Suddenly, the adjective becomes an adverb (the very adverb our third grade teachers told us not to use). This "adverbial accusative" is almost always an "internal accusative" -- that is, an accusative object that renames the action of the verb, even when it is not in any way a cognate accusative. Consider: cano nihil = I sing no song = I don't sing. loquitur multum = he talks much talking = he talks a lot. Thus, you can have a transitive verb ("Im gonna hit your face" = ego faciem tuam icturus sum) with an internal accusative (ego faciem istam multum percussurus sum. = "I'm gonna hit your face big-time" or "I'm gonna smash your face" or "I'm gonna hit your face a lot.").
The accusative case is used to indicate the extent (of space) and the duration (of time): nec unum diem remoratus est = "and he did not wait for one day." tria milia passuum processit = "he advanced three miles." By extension, the accusative is also used to give dimensions (how high, wide and deep something is). This adverbial usage has several possible origins, of which two are sufficient for our purposes. 1) it could be a development of the "goal" function of the accusative: the goal of the journey, Romam venit, being very much the same as the space traversed, decem milia passuum venit. 2) it could equally be an extention of the internal accusative: vitam vixit > longam vitam vixit > multos annos vixit.