The genitive case is a form of a noun that indicates the possessive case of a noun, i.e. the object that the noun is referencing is owning or possessing some other object.
Several languages have genitive cases, including Latin, Greek, German, Russian, Finnish and Sanskrit. Compare nominative case, accusative case, dative case, ablative case, vocative case, ergative case.
In modern English, only pronouns have a distinct genitive declination (my/mine, his/hers/its, our/ours, their/theirs.)
It is a common misconception that English nouns have a genitive case, marked by the possessive " 's " ending. Linguists however have shown that the English possessive is no longer a case at all, but has become a clitic, an independent particle which however is always written and pronounced as part of the preceding word. This can be shown by the following example: "The King of Sparta's wife was called Helen". Now if the English " 's " were a genitive, then the wife would belong to Sparta; but the " 's " attaches not to the word "Sparta" but to the entire phrase "King of Sparta".
That is not to say that the English possessive did not have its origins as a genitive case; but it has developed into being a clitic instead. In Old English, the possessive form of ban (bone) is banes. This developed, later, into the modern English possessive mark of " 's " as in "bone's." The 18th century explanation that the apostrophe might replace a genitive pronoun, as in "the king's horse" being a shortened form of "the king, his horse," is erroneous. Rather, the apostrophe is replacing the "e" from the Old English morphology.
In astronomy, it is important to know the genitive form of the Latin names of constellations, because these are used along with letters of the Greek alphabet to name stars. For example, since the genitive of Gemini is Geminorum, the star Castor, brightest in the constellation Gemini, is named α Geminorum. For more details see Bayer designation.