I had avoided this topic in the past because it is so complex. So, instead of trying to teach you every rule, I'm going to touch on a few, and then give you links to resources that will tackle the rest.
I'm sure you've heard the joke: Commas save lives (Let's eat, Grandma. vs Let's eat Grandma.). Silly as it is, it impresses upon us how much a simple mark of punctuation can drastically alter the meaning of a sentence. The same goes for hyphens.
The error that had me chuckling was this:
"[You're] smart, pretty, funny, sweet, and the hardest working girl I've ever met[.]"
Oh really? 0_o
The line was meant as a compliment, but without a hyphen, it's an insult.
As you've probably deduced, without the hyphen, hard modifies working girl. With the hyphen, hard-working modifies girl.
Though there are exceptions (such as when the first word ends in -ly), the general rule of thumb is: hyphenate two adjectives when they come before the noun they modify and act as a single idea.
Now, if we write That girl is very hard working, we don't need a hyphen, because the adjectives come after the noun they modify.
Compound nouns often use hyphens, too. To determine whether a compound noun is written as one word, two words, or is hyphenated, consult a dictionary. They aren't always written as you think. The examples given by GrammarBook.com are eyewitness, eye shadow, and eye-opener.
The form compound words take can also depend on their function in the sentence. Ex: Break+down is written as one word when used as a noun, but it is written as two words when used as a verb.
Engines break down; they will eventually suffer a breakdown.
For other hyphen uses, see C.S. Lakin's post, in which she gives a link to the Chicago Manual of Style's hyphen rules chart (a downloadable PDF).
That's all for today.
Thanks for visiting. :)