ちがう

[Melchior]

I've sometimes heard 違う used (in entertainment) in place of いいえ.

I know this is the word for different, but I guess it can mean "that's wrong". What does it imply? Is it impolite speech or what?

[nice gaijin]

as a response, chigau/chigaimasu can mean "that's wrong" or "that's not it," or can just be translated as "no." the level of formality is based on the conjugation you use.