A discussion is simply a talk about something. An argument is a talk in which people disagree. Sometimes the argument involves getting angry, sometimes not. The verbs are discuss and argue. Here are some examples:
Hi Bob, I'd like to meet with you today to discuss our plans for the holiday lunch.
We were discussing the surprise party when Cynthia walked in.
Unfortunately, our discussion turned into an argument when Jerry started talking about politics.
Our neighbors argued all night and I could hear every word!
I don't want to argue about it. Let's just forget it!
P.S. Thanks to Bill Watterson for the image from Calvin and Hobbes.