Yes, case systems are part of grammar. Actually, I always call them "grammatical cases," and I think that's the preferred term.
http://en.wikipedia.org/wiki/Grammatical_case ... is always a great place to start.
If you want a nutshell definition, a case is a form of a noun denoting a different function in the sentence. Grammatical case is, in many ways, analogous to verb tense.
For example, let's say "skarn" means person. Then, let's say that any noun ending in "n" is in the subjective case (aka Nominative case), which means that that's the form used when the noun is the subject in a sentence. In many languages, nouns are listed in subjective case, much like how verbs are often listed in the infinitive as a sort of default form.
Then you can say "adding 'ar' changes a noun to accusative case, which makes it a direct object (a noun which receives direct action from the subject via the verb).
So then, if you want to say a sentence like "The tall person hit the short person," the first word for person would be "skarn" since that's the subject of the sentence, and the second word for person would be "skarnar" since that word is the direct object in the sentence.
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I know this is probably confusing, but I tried my best to keep it simple. And don't worry if you can't keep track of all the different cases (there's a link to a list of cases on that wiki page). Most of us here are lucky to remember 5 or 6! XD