Trask writes that possessive apostrophes are the most troublesome of apostrophes, and apostrophes are the most troublesome of punctuation marks. (I think commas must give apostrophes a run for their money.)
The basic rule for possessive forms is that ’s is added to the end:
If the noun ends with s, the same rule usually applies:
The first exception is for a plural noun that already ends in s:
my parents' holiday the ladies' bathroom
These words are not pronounced with an extra iz sound, and so we don’t indicate an extra s in writing.
This brings us to the second exception. If the possessive form of a noun ending in s is not pronounced with an extra s, it only takes an apostrophe:
Aristophanes' plays Bridges' marmalade
The final exception is for pronouns:
Whose is this? The essay is hers
But this does not apply to possessive indefinite or impersonal pronouns:
one's conscience someone's lunch
I have previously covered its as a possessive form, and it is perhaps the pronoun most commonly incorrectly assigned an apostrophe.
Fowler’s Dictionary of Modern English Usage gives a comprehensive breakdown of exceptions and usage in more complicated situations, and I highly recommend it if you are worried about using apostrophes correctly.