What is the rule for the correct use of apostrophes when writing about years? For example, is it the 1970s or the 1970's?
It's the 1970s, but the '70s.
When you have the full date, there's no apostrophe. But when it's only the last two digits, you'd put an apostrophe in front. Think of it like a contraction. There's not apostrophe with "is not," but when you take something out so to speak, you add an apostrophe--"isn't."