What is grammatical case? I'll lead with an example rather than a definition. In English, case is the difference between "I" and "me". In the sentences
- I saw you
- You saw me
both "I" and "me" refer to the same person, but, depending on weather I am doing the verb or the verb is being done to me, English uses a different word. This only happens in personal pronouns in English; we have doers of verbs (I, you, he, it, she, we, they) and the done-tos of verbs (me, you, him, her, it, us, them). For regular nouns, English uses word order to mark who is doing the verb and who it is done to.
- The dog bit the cat
- The cat bit the dog
But some other languages use a morphology system to mark the noun for whether it's the doer or the done-to noun. If it were to use -o for doer and -a for done-to:
- The dog-o bit the cat-a
- The cat-o bit the dog-a
Sometimes this allows languages to have a free word order (or sometimes just less rigid) because the morphology informs the listener who's doing what. Other times the language will still retain a strict word order and a case system, because languages sometimes like to have redundant grammar rules -- In English, you can't say "me saw he" just because you're using case. The word order still matters.
If the language has this kind of case marking, the doer of the verb is called the nominative case and done-to is the accusative case.
Other cases exist. In the sentence "The man gave the bread to the ducks", the man is doing the verb (nominative), the bread is being given (accusative), and the ducks are the recipient of the bread. If you marked the ducks with morphology to indicate they are the recipients, this would be the dative case. English, however, uses a preposition "to".
Locative case is a noun that gives a location of the sentence happening. In the sentence "The man gave the bread to the ducks at the lake", English uses another preposition, "at", to indicate it took place at the lake. But some language will use morphology on "lake" to convey the same concept.
Genitive case sort of exists in English. In "The man's dog bit the cat", the dog is doer and the cat is the done-to. What is the man's role? He is literally the owner of the nominative dog. This is the genetive case, and English marks it with a possessive 's. Broadly, the genetive case draws some relationship between the nominative which includes where the nominative came from, or who the nominative belongs to. This can be literally possession, like "Sarah's pen", where Sarah is the possessor, and thus genitive case. You would also see the genetive case in things "the gunslingers of the West", where "of the West" is the genitive part, except English uses a preposition here and no morphology here. But it's roughly the same as saying "the West's gunslingers".
Vulgar does not (yet) do all the cases that exist. Check out this page for more.