Masculine Feminine Neuter
Article Noun Article Noun Article Noun
Singular Nominative ο λογος η αρχη το εργον
Accusative τον λογον την αρχην το εργον
Genitive του λογου της αρχης του εργου
Dative τω λογω τη αρχη τω εργω
Plural Nominative οι λογοι αι αρχαι τα εργα
Accusative τους λογους τας αρχας τα εργα
Genitive των λογων των αρχων των εργων
Dative τοις λογοις ταις αρχαις τοις εργοις
Masculine Feminine Neuter
Singular Nominative αυτος he αυτη she αυτο it
Accusative αυτον him αυτην her αυτο it
Genitive αυτου his αυτης her αυτου its
Dative αυτω to him αυτη to her αυτω to it
Plural Nominative αυτοι they αυται they αυτα they
Accusative αυτους them αυτας them αυτα them
Genitive αυτων their αυτων their αυτων their
Dative αυτοις to/of them αυταις to/of them αυτοις to/of them

Nominative

The nominative case is the noun the sentence is about.  The one doing the verb.  It is like the subject in English. 

Accusative

The word in the accusitive case is the noun that is being effected by the noun in the nominative case.

Vocative

The noun being spoken to.

Then she came and worshiped Him, saying, "Lord, help me."

ἡ δὲ ἐλθοῦσα προσεκύνει αὐτῷ λέγουσα Κύριε, βοήθει μοι.

Matthew 15:25

Genitive

The noun that has or owns the noun in the nominative case.

ie: ...you should beware of the leaven of the Pharisees and of the Sadducees

...προσέχετε δὲ ἀπὸ τῆς ζύμης τῶν Φαρισαίων καὶ Σαδδουκαίων.

Matthew 16:11

Dative

A noun (and the adjective of it) that is indirectly associated with the nomiative noun (subject of the sentence). In this sentence "they" is the nominative (subject), and you can see that the portion of the sentence "those in the other ship" support the subject but is not the subject.

...and they signaled to their companions, those in the other ship, that they should come and take some of them. And they came and filled both the ships, so-that they began to capsize.

...και κατενευσαν τοις μετοχοις εν τω ετερω πλοιω του ελθοντας συλλαβεσθαι αυτοις: και ηλθαν και επλησαν αμφοτερα τα πλοια ωστε βυθιζεσθαι αυτα

Luke 5:7