Traditionally we talk about apostrophes as being there to show omission and possession. The issue of omission is quite straightforward, and I'll link to some good guides on that. The idea of possession is not quite so simple though, since it does not fully explain the many cases where there is no possession or ownership in any traditional sense.
"That is Karl's book." This is straightforward. I own the book. But then we come across cases like these:
- Scholars’ Walk: do the scholars really own it? No.
- The secretary’s boss: she doesn’t own her boss.
- Karl’s friend: I don’t own my friend.
- 6 weeks’ hard labour: the prisoner doesn’t own the weeks.
I find it much easier to think of apostrophes as representing the 'of' construction. If you can reword it as the "x of x" pattern then use an apostrophe.
- Walk of scholars, therefore Scholars' Walk
- Boss of the secretary, therefore the secretary's boss
- Friend of Karl, therefore Karl's friend
- Two weeks' leave (leave of two weeks, so use an apostrophe)
- BUT two months pregnant (no apostrophe, since you can’t say “pregnant of two months”)
- Citizens Advice (no need for an apostrophe in the organisation name, since it if advice for citizens, not the advice of them - further explanation here, if the link works - if not have a read of the Oxford Guide to Plain English by Martin Cutts).
- "I'm going to John's" = "I'm going to John's [house]" = house of John, so an apostrophe is needed
- Christie's = Christie's [auction house] = auction house of Christie, so an apostrophe is needed
- "Their strengths and weaknesses complemented and cancelled out each other's" = "each other's [weaknesses]" = the weaknesses of each other, so an apostrophe is needed
- An article in The Guardian "If you can't use an apostrophe, you don't know your shit" (and the apostrophes entry in their style guide)
- This Oxford Dictionaries summary
- Wikipedia: omission and 'possession'
I fully expect someone to point out examples where my simple system falls apart!