It's actually very simple. The apostrophe always stands for missing letters:

• "don't" = "do not" (missing letter "o")
• "let's" = "let us" (missing letter "u")
• "B'ham" (on some road signs in UK) = "Birmingham" (missing letters "irming")

You may think, "but hang on, where do possesive nouns come into this? What letters are missing in "John's house"?

Well, I don't know if this is true, but it helps: I've been told that in some unspecified ancient times, people used to say "John his house", which got shortened to "John's house". That theory breaks down on "Mary's", but let's not go too deep into looking for logic in a language, especially English.