“Dialogue” comes from the Greek “dialogos”. “Dia” means through and “logos” means the word. So “dialogue” means doing something through the word, and this makes sense – when we are in dialogue with someone, we are doing something together through speaking or writing.
“Discussion” has the same root as the words “percussion” and “concussion” and means to break things up or smash apart. This seems rather violent, but it does give a good sense of the difference between dialogue and discussion: the latter is primarily concerned with analysis and differing points of view.
But what about “conversation”? Well this has its roots in Latin. “Conversari” meant “keep company with”. We can see how this relates to a modern understanding of “conversation”.
What is interesting, though, is that between the 16th and 19th centuries, “conversation” could also mean sexual intercourse. This meaning is, of course, now obsolete, but it does capture the essence of what is still true about conversation: to have a good conversation you need to connect with the other person. If you are not feeling connected, you're probably not in conversation. And if you are not in conversation, you're quite likely in conflict.