I'm aware that 行かない means "don't go" (hope I'm not wrong).
What I don't understand is the function of で. What's its use in this sentence? (I usually see that character used in です).
いかないで, without any other context, is the short way of saying いかないでください, "please don't go". A Japanese verb ending with the positive て or negative ないで as a single, one-word sentence, is a "please do / please don't" situation:
走らないで(ください)(please) don't run
座らないで(ください)(please) don't sit down
喋らないで(ください)(please) don't talk
来て(ください)(please) come
見て(ください)(please) look
聞いて(ください)(please) listen
omitting ください is often done in casual conversation. It can also give the request a strong feeling when it is omitted.