Apostrophes signal letters that were left out, and they also indicate when a noun belongs to a specific person or thing. The rules are very simple. Memorize them, and you'll know when to use apostrophes - and when you shouldn't!
A "contraction" is just a single word that's been formed from two longer words. For example "shouldn't" really means "should not," but it's easier to use one word instead of two. In fact, the word "not" is the source of many different contractions.
Aren't (are not)
Can't (can not)
Couldn't (could not)
Didn't (did not)
Doesn't (does not)
Don't (do not)
Hadn't (had not)
Hasn't (has not)
Isn't (is not)
Mustn't (must not)
Shouldn't (should not)
Wasn't (was not)
Weren't (were not)
Wouldn't (would not)
Here's a web page that lists all 47 contractions, including 31 more that are identified as "positive contractions." Instead of the word "not", positive contractions are formed from a simple declarative verb, like "is" or "has". Here's some examples of positive contractions.
I'm (I am)
You're (you are)
He'll (he will)
She's (she is)
But sometimes you'll also hear people inventing their own contractions - like an excited boy at Christmas who shouts out "Santa'll be here soon!"
And since apostrophes indicate letters have been left out, there's a special case where they're often used: in abbreviating the name of a decade. Nobody wants to keep saying "the 1960s" every time they're talking about that specific decade. So instead, its consider grammatically correct to use an apostrophe to replace the implied part of the number. You'll even see this in the title of a TV show about the seventies - called "That '70s Show."
But there's another important use for apostrophes: when they're indicating that something belongs to somebody. (That's why it's called a "possessive.") If a teacher has a book, then it's the teacher's book - just add an apostrophe and an S to the end of the word to make that clear. And sometimes even the possessed item can possess something too - for example, the teacher's book's paragraphs!
But what happens if the word already ends in S? It's still acceptable to add another S - for example, if your teachers all follow a syllabus, then it's the teachers's syllabus. But some writers use a different punctuation style, adding just the apostrophe (without the s) - which would make it the teachers' syllabus. Both approaches have their advocates. The Modern Language Association always insists on an apostrophe and an S, while the Chicago Manual of Style says just an apostrophe can be acceptable for words that already end with an S.
There's one exception to this rule: the word its. For example, if a car makes a noise, then obviously that's the car's noise. But if instead of car you said "it," then you would NOT use an apostrophe before the S. (In that example, you would write that "The car was making its noise.") This is done for a good reason, and it's a helpful way to review. Obviously if you were writing a contraction to take the place of "it is," that would get written as "it's." That contraction gets used almost constantly, so to avoid confusion, the English language makes a special exception in the rare cases when instead, you're writing the "possessive" form of it!