If I own something, then the thing that I own is in the 'genetive.' Our nouns don't 'decline' (that is change) according to 'case' (that is what role the noun is playing in the sentence.
So, in English we say 'the dog bites the man,' and we understand what that means without having to change any of the nouns. That is because we express case (largely) through the order of words. If you swap the nouns in that sentence to 'the man bites the dog' you have a different (and far weirder) sentence.
In English, if something is 'of' something, it is in the genitive. 'Is that my apple?' In that sentence 'apple' is in the genitive, because it is the thing being owned.
Or you could say 'That is Mary's book.' This is one of the rare cases in England of a noun's case being shown by the ending (apostrophe, s.)
In English we know if something is a genitive either because it is 'of' somebody, or because it has apostrophe s after it.
In Irish the genitive is formed differently. And I understand we will get to that more in det
You are, of course, right. It has been twenty years since I studied Anglo Saxon, but that is no excuse for me muddling up the genitive case like that. Anglo Saxon was heavily inflected, compared to modern, or even middle English.
The apostrophe 's' is our modern version of the Anglo Saxon way of demonstrating ownership, in which the noun doing the 'owning' ended in an 'es.' A dog. A doges owner. A child. The childes mother. A king, a kinges subject. (of course the nouns I'm describing were not our modern English version, but that's the basic concept.)
Over time the vowel was 'swallowed' (fairly typical of English pronunciation norms) and became 's.' Because the 'e' was missing from the 'es', we ended up with an apostrophe to mark its absence, but have never dropped the apostrophe, even though it is nigh on a thousand years since anyone last used it. (Irregular pronouns don't have the apostrophe 's.' (Which is why I am able to type 'I don't like its smell, apostropheless.)
Here (lifted from Wikepedia, are the inflected nouns.
The nominative case indicated the subject of the sentence, for example: se cyning means 'the king'. It was also used for direct address. Adjectives in the predicate (qualifying a noun on the other side of 'to be') were also in the nominative. The accusative case indicated the direct object of the sentence, for example: Æþelbald lufode þone cyning means "Æþelbald loved the king", where Æþelbald is the subject and the king is the object. Already the accusative had begun to merge with the nominative; it was never distinguished in the plural, or in a neuter noun. The genitive case indicated possession, for example: the þæs cyninges scip is "the ship of the king" or "the king's ship". It also indicated partitive nouns. The dative case indicated the indirect object of the sentence; To whom or for whom the object was meant. For example: hringas þæm cyninge means "rings for the king" or "rings to the king". Here, the word cyning is in its dative form: cyninge. There were also several verbs that took direct objects in the dative. The instrumental case indicated an instrument used to achieve something, for example: lifde sweorde, "he lived by the sword", where sweorde is the instrumental form of sweord. During the Old English period, the instrumental was falling out of use, having largely merged with the dative. Only pronouns and strong adjectives retained separate forms for the instrumental.
In the politest way possible, I just feel I ought to correct this here slightly: the genetive is the 'owner', and the 'owned' noun can be a subject or object (or uncommonly another genitive, in a phrase like 'my mother's brother's uncle').
So, in above examples like 'A doges owner', the word 'doges' is in the genetive. Everything else was well put. Just got that bit switched.