The genitive relation consists of a noun phrase (the minimum noun phrase in most languages being a single noun) that modifies another noun phrase. As described by grammar books, a genitive phrase is a noun phrase that specifies or describes another noun phrase, with the result being that the genitive phrase helps to answer the question "which?" with regard to the modified noun phrase; e.g. "the girl next door's dog" (genitive in italic) answers "which dog?", "music of Starcraft" answers "which music?", etc.
As it essentially contains all manner of modifying phrases, clearly the genitive is an extremely broad relation, and there are quite a few more-specific relations that fall under the umbrella of "genitive". A semi-exhaustive listing of the various types of relations the genitive can express:
- Physical possession: "my computer"
- Abstract possession: "your happiness"
- Relational: "his wife"
- Compositional: "bottle of water", "box of nails"
- Quantitative: "half of them"
- Subjective: "her snoring"
- Objective: "its destruction"
- Purpose: "sledgehammer of castration"
- Location: "citizens of America"
- Origin: "Kazuhiro Sasaki of Japan" (for those who don't know, he's a Japanese baseball player that was recruited to play professionally in America)
- Affiliation: "Apple's Steve Jobs"
- Attributive: "thing of beauty"
- Topical: "Of Mice and Men"
- Appositional/classifying: "President Obama"
One very good reason for a language to have more than a single mechanism of representing the genitive is that any single such mechanism would necessarily be ambiguous. Take, for example, the English phrase "betrayal of Illidan"; with this phrase it is unclear whether the genitive is subjective or objective - that is, whether Illidan is the betrayer or the one betrayed. Other such examples can be invented based on other uses of the genitive.
Of course it's not all bad; if there were no ambiguity in language, puns would be impossible. One real-life example of genitive ambiguity comes from IRC. jfroy says 'Snow Leopard meeting' in the sense of 'meeting about [Apple] Snow Leopard', to which I reply, interpreting that as 'meeting with a snow leopard':
<@jfroy> Snow Leopard meeting over. Ow.
<enma_hinobara>Next time don't poke it
<enma_hinobara>It's not Kaity
The Genitive in English
It's arguable whether English still has a genitive case; if it does, it only exists for (some) pronouns, where it indicates a purely possessive relation (which is why it might better be called the possessive case). It does, however, have three different mechanisms for expressing genitive relations in general - the periphrastic genitive, the analytic/agglutinative genitive (both are my own terms, so don't expect to find them in a dictionary), and the possessive.
The periphrastic genitive renders the genitive phrase as a prepositional phrase with the preposition 'of', e.g "fog of war". This is the "true" genitive in English, in that it is the mechanism that can express nearly every possible genitive phrase (although some phrases may sound better using one of the other mechanisms), while the others are much more restricted in use.
The analytic/agglutinative genitive renders the genitive relation purely by shoving two (or more) nouns together, e.g. "milk carton" (compare to "carton of milk"), "wood chips" ("chips of wood"), etc. I call it the analytic genitive because the genitive relation is expressed purely analytically - through word order, rather than word form. I call it the agglutinative genitive because in more synthetic relatives of English, such as German or Old English, the modifying word(s) would be agglutinated with the modified word to form a single large word (this can sometimes produce very long compound words); for example, "girl next door" would be "Nachbarmädchen" in German (literally "neighborgirl"). In English, this type of genitive is restricted to certain types of relations, and is especially used for classification.
Finally, English has a special means of expressing possessive relations, a subset of genitive relations (it's unclear exactly how this form originated; some argue that it's an evolution of the genitive case, while others are more skeptical of that conclusion). Note here that English has a rather broad concept of possession, and as such the possessive can be used with some relations that aren't strictly possessive in nature.
The Genitive in Latin
As a relative of English, the genitive in Latin is much the same, in that genitive relations can be expressed via prepositional phrases (which are substantially similar to their English equivalents) or analytically; however, Latin also has an actual genitive case (a result of which is that it does not need a possessive structure as English does), which is used very commonly. A few examples of the genitive case in Latin:
- "agricolae [farmer, genitive] filia [daughter, nominative]": "farmer's daughter"
- "horum [these, genitive] omnium [all, genitive]": "of all these/of all of these"
- "vir [man, nominative] magnae [great, genitive] virtutis [courage, genitive]": "man of great courage"
- "fossa [ditch, nominative] decem  pedum [foot, genitive]": "ditch of ten feet/ten-foot ditch"
- "Rex [king, nominative] belli [war, genitive] cupidus [desirous, nominative] est [is]": "The king is desirous of war"
Finally, As Japanese does not have true case, like English it uses non-case-based structures for constructing the genitive. Japanese has two genitive-marking particles - 'no' and 'na', which differ only in whether the genitive phrase is abstract (e.g. 'baka' - 'stupidity') or concrete (e.g. 'shakunetsu' - 'scorching heat') - used in the periphrastic genitive, as well as the agglutinative/analytic genitive (the genitive may or may not actually be agglutinated). However, Japanese also shows off several uses of the genitive that English and Latin do not.
Consider the rough definition of the genitive - something which specifies or defines a noun phrase; where have we heard this definition before? Well, for one, it's the definition of adjectives. In fact, a language can get by just fine with almost no true adjectives (e.g. the kind we have in English), instead opting to use one or more of the three alternate methods I talked about previously. One of these methods is to use the genitive with descriptive nouns, a method that is used extensively in Japanese and Caia (in fact, it's the primary method in Caia). A couple example from previously used words would be "baka na inu": "stupid dog" (literally "dog of/with stupidity") and"shakunetsu no koi": "scorching love" (literally "love of scorching heat").
The other structure this definition matches is apposition; that is, a restatement of something previously said for the purpose of definition or clarification, such as the italicized part in "Julius, son of Ambrose". In a previous post on the predicative, I explained that in Indo-European languages such phrases are in the same case as the phrase they restate, but that we could imagine other reasonable ways of expressing them. In Japanese such phrases may be expressed either by the analytic genitive (e.g. "kemono [beast] no sousha [player, generally of an instrument] Erin": "Erin the beast player") or the periphrastic genitive (e.g. "Hamelin no violin hiki [player]": "Hamelin the violinist" - literally "violinist of Hamelin").
Finally, some languages, such as Japanese and Caia, allow other relational phrases to be put into the genitive, where the genitive particle indicates the modification of a noun phrase, and the other relational particle specifies the precise relation of the genitive phrase (you might say the genitive particle applies top-down, while the other relational particle applies bottom-up). Some examples of this in Japanese:
- "ano [that] group to [with] no kankei [relationship]": "relationship with that group" (literally "relationship of with that group")
- "ano kaisha [company] to no keiyaku [contract]": "contract with that company"
- "Asia kara [from] no ryuugakusei [exchange student]": "exchange student from Asia" (literally "exchange student of from Asia")
- "Hokkaido kara no omiyage [souvenir]": "souvenir from Hokkaido"
- "America ye [towards/to] no monkowohiraku [opening the door; this is rendered as a noun in this phrase, not a verb, in this example]": "opening the door to America" (literally "opening the door of to America")
- "anata [you] ye no tegami [letter]": "letter to/for you"