Possesive case is a case that exists in some languages used for possession. It is not the same as the genitive case, though the two have proximal meanings in many languages.
There are many types of possession, but a common distinction is alienable versus inalienable possession. Alienability refers to the ability to dissociate something from its parent -- in this case, a quality from its owner.
When something is inalienably possessed, it is usually an attribute: for example, John's big nose is inalienably possessed, because it cannot (without surgery) be removed from John -- it's simply a quality he has. In contrast, 'my briefcase' is alienably possessed -- it can be separated from me.
Many languages make this distinction in some way. Saying something like 'I have my dad's big nose' with the latter noun-phrase marked inalienable would imply some sort of genetic inheritance; marked alienable, it would imply that you had cut off your father's nose or somesuch and were actually in physical possession of it.
English does not have a grammatical facility to make such distinctions.
The term 'possessive case' is often used to refer to the "'s" morpheme, which is suffixed onto many nouns in english to denote 'possession by'. This usage is not strictly correct -- this affix is actually a clitic. By descent, however, the English usage does stem from a case ending, Old English -es. See genitive case for details.
Compare accusative case, nominative case, dative case, ergative case, genitive case, vocative case, ablative case.
