Em dashes are what the non-copyediting world call dashes. Constructed from three hyphens, they are easy to use and easy to overuse.
1) Use an em dash when your sentence is going in a dramatically different direction from itself. “They say I have ADD, but they don’t understand—oh look! There’s a chicken.”
2) Use an em dash to elaborate on a point. “Desdemona hated to do her homework—especially algebra.”
3) Use an em dash as a strong comma, or to present an aside “Desdemona had to do her math homework—algebra, no less—before she could go out for ice cream.”
Long story short: The em dash is all about interruption and directional changes in sentences. Do not overuse them. They become a crutch, a way to make language more expressive or to control the flow or pace. The em dash is like perfume. A little perfume is gorgeous and alluring. A lot of perfume is off-putting and gross.
Those of you who are writing dialogue can use them more often than the rest of us.
Questions? Comments? Need a treatise on how Emily Dickinson used the dash? Other punctuation marks you’ve never understood? Do you hate me now?