I've been thinking about apostrophes recently, following Cambridge Council's dropping of them, then their reinstatement following public pressure (including my digging). I've been asked questions about apostrophes, and when they should be used. I thought I would do a bit more public service and explain the rule I work by.

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.
So the simplified description 'possession' obscures as much as it reveals, and isn't the best way to think of aspostrophes.

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
This also works with cases that normally cause problems:
  • 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 is 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). 
Bear in mind the complication when words are missed out, but have to be assumed, and therefore an apostrophe is still required.
  • "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
None of this is rocket science, and I'm not claiming to have made it up, it is just the system I find simplest to adopt. If you want to know more about apostrophes then I recommend the fuller articles here:
I fully expect someone to point out examples where my simple system falls apart!