Nominative, accusative, dative, genitive, what exactly are these related to grammar?
Nominative, accusative, dative and genitive are all grammatical cases.
They vary in function in different languages.
Here is what they look like in English:
nominative - subject
e.g. I ate some pie.
Here, I would be in the nominative since it is I that was doing the verb (eating).
accusative - direct object
e.g. Do you have money?
Here, money would be in the accusative since it is the pet that the verb is being done to- the pet is owned by the person.
dative - indirect object
e.g. I bought a horse for my friend.
Here, the dative is 'for my friend'. The reason why this is not in the accusative is that I am buying a horse (the direct object in this sentence), rather than my friend.
genitive - possession
e.g. The boy's balloon is gone.
This case is easier to notice since the word itself usually changes. "'s" is added to the nominative word, so 'the boy -> the boy's'. Personal pronouns also change (e.g.he, she, it -> his, her, its).
It is worth saying that cases only affect nouns in English, though in other languages they can affect adjectives.