The accusative case of a noun is, generally, the case used to mark the direct object of a verb. The same case is used in many languages for the objects of (some or all) prepositions.
Modern English, which lacks declension in its nouns, still has an explicitly marked accusative case in a few pronouns as a remnant of Old English, an earlier declined form of the language. "Whom" is the accusative case of "who"; "him" is the accusative case of "he"; and "her" is the accusative case of "she". These words also serve as the dative case pronouns in English and could arguably be classified in the oblique case instead. Most modern English grammarians feel that due to the lack of declension except in a few pronouns, where accusative and dative have been merged, that making case distinctions in English is no longer relevant, and frequently employ the term objective instead (see Declension in English).
Example
I see the car. Here, the car is the direct object of the verb "see". In English, which has mostly lost the case system, the definite article and noun — "the car" — remain in the same form regardless of the grammatical role played by the words. One can correctly use "the car" as the subject of a sentence also: "The car is parked here."
In a declined language, the morphology of the article and/or noun changes in some way according to the grammatical role played by the noun in a given sentence. For example, in German, one possible translation of "the car" is der Wagen. This is the form in nominative case, used for the subject of a sentence. If this article/noun pair is used as the object of a verb, it (usually) changes to the accusative case, which entails an article shift in German — Ich sehe den Wagen. In German, masculine nouns change their definite article from der to den in accusative case.
(in English and some other languages) a case specialized for the use of a form as the object of a transitive verb or of a preposition, as him in The boy hit him, or me in He comes to me with his troubles.
(in case grammar) pertaining to the semantic role of a noun phrase that denotes something undergoing a change of state or bearing a neutral relation to the verb, as the rock in The rock moved or in The child threw the rock.
Of, relating to, or being the case of a noun or pronoun that serves as the object of a verb.
Thus in the English sentence "He helped him," him is in the accusative (or, as it is sometimes called, objective) case, he in the nominative.
Grammatically, Czech has seven cases (nominative, genitive, dative, accusative, locative, instrumental, and vocative) for nouns, pronouns, and adjectives.
As in Czech, the nouns, pronouns, and adjectives have seven cases (nominative, genitive, dative, accusative, vocative, instrumental, and locative).