What does it mean to "flesh out" a character? How is this done?
Different characters (just like real people) look different and act differently; the have different likes, dislikes, habits, aspirations, and fears.
The sheet Madison linked to:
This character chart (or at least parts of it) always helps me...
can help you differentiate characters by building them with different traits.
OK, but
why would you want to do that? The short answer is that it makes your characters three-dimensional; it gives them their own personality.
The long answer is that is makes your character--and your story--interesting.
Imagine a character...oh...walking to the store to but dogfood, for example. She's high-strung, so she walks fast, but is careful where her heel lands, because she doesn't want to twist her ankle again. She's got long hair that she likes to wear loose, and she tends to unconsciously brush it out of her eyes, particularly on windy days such as this. She likes red, although it really isn't her color, and the blazing fire-red jacket she wears is an eye-catcher, but it makes her look pale. The off-red pumps and small purse don't quite match; she's not as fashionable as she thinks she is. Perhaps she smokes, and stops to light up--the cigarettes are in her jacket, rather than her purse, because she's trying to force herself to quit. Her fingernails are long (and red, of course), but they're press-ons. She's worried about her father, who seems to be getting Alzheimer's, so her thoughts aren't really on the walk or the dogfood. She walks past the house where her best friend in third grade once lived.
And so forth and so on. Now you certainly
don't want to dump all this information on the reader for no particular reason, but
knowing about your character allows you to add life. You don't have to actually
say she likes red, but because of that, you know more about her wardrobe. You don't actually have to
say she's trying to quit smoking, but that fact will alter her behavior. You don't actually have to
say she like to wear her hair loose, but what happens when it's breezy?