Ellipses are those three little dots ... They are used to indicate an omission or something is missing from a long direct quotation, if for example you are only using part of a quote and leaving off a few words. The ellipse goes at the point of omission.
More often in novel form, the ellipse is used to indicate hesitation or loss of concentration in a character's dialogue. Or perhaps a character trails off and doesn't finish a sentence of dialogue or thought. The sentence will then end with ...
I usually see these handled with a space before and after, such as:
"I wish I could tell you about ... I just can't."
"Look at his abs. They are so dreamy and ..."
Why did I ever start this post? It's so hard to think of examples when I should be cleaning. I wish ...
"Um ... where do I go from here?"
An em dash, on the other hand, indicates an insertion that isn't quite on topic. An aside by the character or narrator. It can also be used in dialogue when a character is interrupted while speaking. It is not a hyphen, but two hyphens typed side by side with no spaces around them. Word will automatically connect them into a longer em dash when done correctly.
“I love eating tacos—the extra-loaded kind from that quaint Mexican place—when I’m studying for finals.”
“Is that my—”
“Yes, it’s your dress. I borrowed it.”
“Today, I found the best sale—all the best sales happen this time of year—on shoes.”
An em dash is also used when inserting a tag in the middle of dialogue.
“My cat got”—he dashed at his eyes to wipe away tears—“run over by a truck.”
(No cats were harmed in the making of this post.)
How about some examples of how you've used them in the comments? Here's your chance to be fun and creative. It's not so easy to make them up on the spur of the moment, is it?
Edit: Em dashes are becoming more popular and are often used in place of semi-colons now. My Harper Voyager editor prefers this.