Proper apostrophe use

The apostrophe has two, and only two, uses: to show possession and to indicate the omission of letters or numbers. To further illustrate this point, let us examine some of the rules that dictate when apostrophes should be used and where they should be placed in a word.
Common rules for apostrophe use

Possessive common nouns are common
nouns or pronouns that own other nouns. Apostrophes are used to indicate this possession in the following ways:
If the noun does not end in -s (in most cases this means it is singular), add -'s.
Here are two examples:
The bike's handlebars were bent in the crash.
The boy's sister traveled by bus to meet us.
If the noun is singular and ends in -s, add -'s, as in the following examples:
My boss's job at the bank was eliminated due to budget cuts.
The class's average grade was impressive.
If the noun is plural and ends in -s, add only an apostrophe.
The clowns' shoes protruded from the window of the Volkswagen.
Both bananas' peels had turned brown.
If the noun is plural and does not end in -s, add -'s.
The children's play received a standing ovation.
The geese's precise formation in the sky impressed the pedestrians.
Some words or phrases are awkward to pronounce when the apostrophe is added ("geese's precise formation," for example). An author always has the option of rewriting the sentence to avoid this problem ("The precise formation of the geese...").
If multiple nouns jointly own another noun, use an apostrophe only on the final noun listed. In this sentence, one car belongs to both the man and the woman.
The man and woman's car was badly damaged.
If multiple nouns each possess another noun individually, each noun should have an apostrophe. In this sentence, there are two separate motivations, each owned by a different person.
The student's and the teacher's motivations were in conflict.
If a compound noun owns another noun, add the apostrophe only to the last element.
My sister-in-law's love of shopping knows no limits.
The president-elect's agenda proposed no major policy changes.
If an indefinite pronoun (a noun that refers to no specific person or thing) owns a noun, add -'s.
Someone's car is parked in the loading zone.
Does anybody's key fit this lock?
Proper nouns and apostrophes
Possessive proper nouns are the capitalized names of specific persons, places, or things. We recommend following the same rules for apostrophe use on proper nouns as you would on common nouns. For example:
If the name does not end in -s, add -'s.
Sally's hair was blond and curly.
The Boston Globe's editorial page is popular.
If the name ends in -s and the pronunciation is not terribly awkward, add -'s.
Robert Burns's poetry is difficult to understand.
Charles Dickens's novels contain an astonishing number of characters.
There are a few exceptions to this rule, of course.

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Culled from scribendi