The lowly apostrophe has been taking a beating, lately, and virtually everyone has blood on his hands. Here is a quick run-down on rules for using this simple, useful piece of common punctuation.

Missing Letters
One main purpose of the apostrophe is to recognize the removal of certain letters from a word or pair of words, resulting in a new pronunciation. Examples:

"they have" becomes "they've"
"we are" becomes "we're"
"three of the clock" becomes "three o'clock"


Possession
The most common use of the apostrophe is to show ownership. It is a peculiarity of the English language that we use a punctuation mark to show possession in more than half of cases.

But, creating possessives can create a lot of special circumstances, and even today, our language is evolving, giving rise to new, more lenient methods of creating the possessive.

1. Singular subject not ending in s -
   Simply add 's to the end.
   Example: Bob's Car


2. Singular subject ending in s -
   Always add 's to the end, except when the next word begins with "s", in which case, you only add the apostrophe
   Example: The waitress's apron. The waitress' sandals.


3. Plural subject not ending in s -
   Simply add 's to the end.
   Example: Women's liberation.


4. Plural subject ending in s -
   Only add the apostrophe.
   Example: The students' grades.

There are, of course, words which do not exactly fit one of these four categories, but adherence to these rules will be correct in the vast majority of cases. To be right every time, I recommend becoming intimately familiar with one of the recognized guides on the subject, either the AP Stylebook or the MLA Handbook.

Strangely, many people mistakenly use the apostrophe for the purpose of pluralization, which is never correct. Here are some examples:
Those badger's are vicious. The cop's pulled me over.

This usage is absolutely never correct. If you are making the singular into a plural, over 90% of the time you just add an "s" to the end, and it's wrong 100% of the time to use an apostrophe.