I. The Genitive:

(I.1) You use an apostrophe with a possessive singular noun, whether it’s a proper
name, e.g.

“Adrian’s fanaticism about apostrophes is tragic.”

or not, e.g.

“The computer’s battery is dead.”

(I.2) You even use an apostrophe with possessive plural nouns, e.g.

“Artists’ books are illuminating.”
“Members’ privileges have been revoked.”

(I.3) You always use an apostrophe with possessive proper names that end with an s, whether you hear – and therefore see – the possessive s, e.g.

“You must read Marcus’s new book.”

or whether it is absent in sound and therefore in print, e.g.

“You must study Socrates’ Apology carefully.”¹

However,

“You must read John Rawls Theory of Justice.”

is never correct in English.

(I.4) You use an apostrophe with the impersonal possessive pronoun, e.g.

“It is not always easy to find one’s bearings.”

(I.5) You NEVER use an apostrophe with a third-person possessive pronoun. E.g.

“The mistake was hers” is right;
“The mistake was her’s” is wrong;
“It has its own source of power” is right;
“It has it’s own source of power” is wrong.

(I.6) You NEVER, EVER use an apostrophe to designate simple plurality, EVER. So, the following are wrong:

“the 1960’s”
“CD’s”
“members’ privilege’s”

and the following are right:

“the 1960s”
“CDs”
“members’ privileges”

so please ignore the New York Times’ practice here, because they’re wrong and it makes them look illiterate, like Riddley Walker.

¹Thanks to Jennifer Higgie of Frieze for this sensible rule.

II. Contractions:

(II.1) Aside from (I), apostrophes are used ONLY for contractions, e.g.

“It is time to get up” becomes “It’s time to get up;”
“He has not called” becomes “He hasn’t called;”
“There is a flea in my soup” becomes “There’s a flea in my soup;”
“The 1960s were great years” becomes “The ’60s were great years;”
“I like your new hairdo” becomes “I like your new ’do;”

and so on.