The nominative case is the noun the sentence is about. The one doing the verb. It is like the subject in English.
The word in the accusitive case is the noun that is being effected by the noun in the nominative case.
The noun being spoken to.
Then she came and worshiped Him, saying, "Lord, help me."
ἡ δὲ ἐλθοῦσα προσεκύνει αὐτῷ λέγουσα Κύριε, βοήθει μοι.
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.
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