Below are the key rules for using the apostrophe (').
These rules are those which apply in modern British English (as taught in High Schools and Universities throughout the Commonwealth).
When you add an 's' to make a plural, it NEVER takes an apostrophe. Never. There are no exceptions.
The same is true for plurals of abbreviations - it is A.T.M.s,
not A.T.M.'s and G.P.s not G.P.'s.
contracted words - words formed by joining two words together and missing
out some letters - the apostrophe takes the place of the missing letters
becomes We've, and so on.
Apostrophes are used to make a noun possessive
- i.e. to show the ownership of an item.
a singular noun, (such as John, horse, New Zealand), an apostrophe and an
s are added at the end of the word to indicate ownership:- John's
coat, the horse's tail, New Zealand's beautiful
scenery. The exception to this is singular nouns which end in the letter
s. For possessives, these are treated in the same way as plural nouns.
possessive form of plural nouns and singular nouns ending in s
(such as boys, dogs, soldiers, princess) is indicated by placing an apostrophe
at the end of the word, but no s after the apostrophe:- the boys'
boots, the dogs' collars, the soldiers' rations, the princess'
possessive form of a pronoun does not take an apostrophe at all but
instead is an entirely separate word, as set out below:
alternatives are given, which you use will depend on the structure of the
Me - My/Mine
You - Your/Yours
He - His
She - Her/Hers
It - Its
We - Our/Ours
They - Their/Theirs
Who - Whose
This is my/your/her/our/their house or
This house is mine/yours/hers/ours/theirs.
There are two very common mistakes in apostrophe use, and both come from not following the rules set out above:
Its, It's and Its': The possessive form of the word it is its, without an
apostrophe anywhere, because it is a pronoun and follows the special
rules for possessive pronouns. It's can only mean it is, or it has
following the rules for contractions, and depending on context, and its' doesn't exist at all, since the
plural of it is they.
Whose and who's: Once again, who is a pronoun, and following the rules for
possessive pronouns, the correct possessive is whose. The word who's
is a contraction of who is or who has.
If you use these rules you will never be using the apostrophe incorrectly, since although U.S. English is, as Gorgonzola points out below, less prescriptive, there is no situation where applying the rules above would be positively wrong, Strunk and White notwitstanding.