A niece is your brother or sister's daughter. It's as simple as that. Here's an analogy: niece is to female, as nephew is to male. Get it?
Your parents' brothers and sisters are your uncles and aunts. If you are female, you are their niece. If you are male, you are their nephew. Both niece and nephew come from the Latin word nepotem. One nice thing about the spelling of niece is that it adheres to the "i" before "e" except after "c" rule. Whew!