The case of a noun or pronoun (and, in some languages, adjective) tells us how it is grammatically related to other words in a sentence. Grammarians distinguish five main cases in Greek (Latin, e.g., has six, while Sanskrit has eight):
1. The nominative (nom) indicates that the noun is the subject of the verb; e.g., "Mary is resting."
2. The vocative (voc) is the case of address; e.g., "Master, I am coming."
3. The accusative (acc) indicates that the noun is the direct object of the verb; e.g., "They stoned Stephen."
4. The dative (dat) points to the indirect object of the verb; e.g., "He gave Peter the key," or "Jesus passed the bread to Judas."
5. Genitive (gen) is the case of possession; e.g., "This is Paul's manuscript," or "That is the mother of the beggar."
The ablative is sometimes raised in discussion of the Greek text, though it is more relevant to Latin. The form of a the noun, pronoun, or adjective in the ablative indicates by whom or what something is done, or where something comes from. A Greek word in the genitive ("of his wound") in a clause like "he died of the wound," may be understood as ablative, i.e., the wound was the cause of his death.
The ability to identify the case of a word is a fundamental competence for exegesis, especially when dealing with the Greek text, where word order is not important.
The term 'case' is derived from the Latin word casus for 'falling,' because the substantive (nouns) was understood "as falling in a certain relationship to the rest of the sentence" (Chamberlain) with the nominative falling in a direct relationship and the other cases indirectly. It is from this idea that we speak of "inclining" the nouns and the other parts of speech qualifying them (adjectives, pronouns, definite article) and the patterns of inflection for the various cases "declension" of which there are three, depending on the word endings.
Note also that the vocative is, properly speaking, not a case but is treated by grammarians as such.
Resources
Murray Vasser, "Intro to Greek 8: Nominative & Accusative."
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Murray Vasser, "Intro to Greek 9: Genitive & Dative."
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