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Possessive marker, indicating than an object belongs to the noun or noun phrase bearing the marker.
The cat bit the dog's tail and ran.
The cat bit the dog with the shaggy fur's tail and ran.
May be used to mean "the house/place/establishment of" when added to a name.
We're going to Luigi's for dinner tonight. (that is, Luigi's house or Luigi's restaurant)
Contraction of is.
The dog's running after me!
Contraction of has.
The dog's been chasing the mail carrier again.
(informal) Contraction of does.
What's he do for a living?

Usage of the possessive marker with words ending in s varies. The final s is dropped after regular plurals.
- the dog's tail but the dogs' tails.

and may or may not be dropped after words ending in s, depending on one's pronunciation (see the rule of thumb below)

Irregular plurals with endings other than s always take 's.
- the children's voices

Possessives can generally be recast using of the, and this may be advisable if the contraction seems awkward or ambiguous (which is often the case in speech).
- the tails of the dogs.

Traditionally, the possessive of the biblical name Jesus is written without a final s.
- Jesus' disciples

This may or may not apply to the Spanish given name Jesus.
- This is Jesus Ramirez, and this is Jesus's wife.

When referring to things possessed by two people, the strictly correct form is to form the possessive of both names, as in "Jack's and Jill's pails". It is common to treat the pair of names as a noun phrase and to form its possessive instead, using only one 's, as in "Jack and Jill's pails".

Rule of thumb

An excellent rule of thumb for determining whether to add 's or an apostrophe alone is the following: If you add an s to the noun when pronouncing the possessive, add 's when writing the possessive; otherwise add only an apostrophe. Examples:
- the possessive of dog is pronounced dogs, so add 's to dog to get dog's;
- the possessive of dogs is pronounced the same way, so add just an apostrophe to get dogs' ;
- the possessive of James can be pronounced either Jameses or James, so add either 's or just an apostrophe to get James's or James' , according to your pronunciation;
- the possessive of Joneses (meaning the people making up the Jones family) is pronounced Joneses, so add just an apostrophe to get Joneses' .

possessive marker

French: de (after the thing owned and before the owner)
the dog's tail - la queue du chien
Italian: di (after the thing owned and before the owner)
the dog's tail - la coda del cane

the house/place/establishment of

contraction of "is" See is

contraction of "has" See has

contraction of "does" See does

Used to form the plurals of numerals, letters, some abbreviations and some nouns, usually because the omission of an apostrophe would make the meaning unclear or ambiguous.
There are four 3's in my phone number."
"Banana" has three a's and one b. (apostrophe s used so that the plural of "a" is not confused with the word "as")
You can buy CD's in that shop.
These are the do's and don'ts. (apostrophe s used as "dos" may be misread)
(incorrectly) Used to form the plural of nouns that correctly take just an s in the plural. See greengrocer's apostrophe.
Apple's 50p a pound

The use of 's to form plurals is to be avoided except when the meaning would be unclear. The following are preferable:
- 3s, CDs
However, do's is commonly seen, and the plurals of letters, especially of lower-case "a" and "i", are best written with an apostrophe to avoid confusion.
The use of the apostrophe in any plural (as in "apple's") — the so-called "greengrocer's apostrophe" — is incorrect.

See -s