Really, there are three main reasons why an apostrophe is used:
1. to form the possessive of a noun
2. to show the omission of letters
3. to indicate certain plurals of letters, mainly done for clarification purposes
1. The father of the child will become: The child's father -> possessive singular
The combined stupidity of two nations: The two nations' combined stupidity -> possessive plural (note that the s comes BEFORE the apostrophe)
2. It is cold becomes: It's cold (note that it's and its are two entirely different words)
They are rather crazy: They're rather crazy
Do not mock me for my grammar obsession: Don't mock me for my grammar obsession.
Who is responsible: Who's responsible (note that who's and whose have two very different meanings)
I could have screamed: I could've screamed (Please don't get into the habit of replacing the 've with 'of,' that is very, very wrong!)
3. Mind your p's and q's is a fairly common term, possibly stemming from the days when manners were still taught to smaller children, and the mothers would simply shorten the phrase "Mind your pleases and thank yous."
More on this on the internet About Grammar and a lot more detail here