The possessive 's, also known as the Saxon genitive or genitive 's, refers to the use of an apostrophe (which may or may not be followed by the letter "s", depending on the context) to show that the following noun or noun phrase is associated in some way to the preceding noun, i.e. Mary's house = the house of Mary; goat's cheese = cheese made from the milk of a goat and Smith's outstanding lifetime achievements.
While intuitive to native English speakers, many students have difficulty in grasping its use and often try to put the definite article in front of proper nouns, etc., as follows: *. That said, many native users of English also make mistakes, the infamous "greengocer's apostrophe" being a case in point, as is the relatively frequent confusion between it’s and its  and "your" and "you're".
To further complicate matters, some English teachers are guilty of misleading their students into believing that the possessive 's is only to be used when referring to people, organisations and/or animals. This is simply not so, as the following examples show: a reward for a lifetime’s work and yesterday's weather....
- We use the possessive 's to talk about possession, relationship, physical features, etc. We use it to talk about a noun which “belongs” to a person, a country, organisation, etc.: John’s hair; Anne’s flat; the company's marketing strategy; Spain’s unemployment figures
- we also use it to talk about something which is used by a person or animal: There are two birds’ nests in that tree; a child’s bicycle;
- It can also be used for products from living animals: cow’s milk; lamb’s wool;
- It is often used to say how long things last: three hours’ journey; twenty minutes’ delay;
- It can indicate the origin of something in a proper name: Murphy's law;
1. singular noun + ’s: Peter’s friends gave him a great present;
2. plural noun + s’: The athletes’ representative negotiated the deal;
3. irregular plural + ’s: The women’s husbands were watching football on TV;
4. We can add ’s or s’ to a whole phrase: Henry the Eighth’s six wives; Anne and John’s house in the country;
While the difference between plural possessives can be shown in written English it can be ambiguous in the spoken language. For example the pronunciation of (2) would be the same if it were:
- The athlete's representative ...
- The athletes’ representative ...
In practice, however, context usually makes things clear.
 Lack of consensus
There's no consensus on how to use it with surnames ending in "s", for example "Davis". By way of example, until its 15th edition (2003), the Chicago Manual of Style recommended Davis', but in its 16th edition (2010), recommends Davis's. On the other hand, Oxford Dictionaries recommends the former.
Especial care must be taken in the use of the possessive apostrophe in titles. In the same way that we don't say or write *, we don't say or write *. The correct way is "Shakespeare's Tempest". We can also modify it and say "Shakespeare's last play The Tempest".
- Homer's Iliad - not: *
- "Possessive apostrophe" World Wide Words
-  Oxford Dictionaries. Retrieved 30th September 2012.
- "Apostrophe" Oxford Dictionaries. Retrieved 30th September 2012.
- "Possessives and Attributives" The Chicago Manual of Style. Retrieved 30th September 2012.
- Trask, R. L. Mind the Gaffe (2001), ISBN 0-14-051476-7