An argument, in the logical sense, is a set of premises offered in support of a conclusion. That's it. It isn't a fight, and it isn't a strongly-held opinion stated loudly. If someone says 'immigration is good' and someone else says 'immigration is bad,' no arguments have been made — just two conclusions with no supporting structure.
Once you can see the structure, you get two independent ways to disagree, and confusing them wastes enormous time. You can challenge whether the premises are true, or whether the conclusion actually follows from them even if they are. An argument can be perfectly logical and still worthless because a premise is false; an argument can have all-true premises and still fail because the conclusion doesn't follow. The single most useful habit in reasoning is asking, out loud: what exactly is the claim, and what exactly is supposed to support it?
Aristotle's great insight, roughly 2,300 years ago, was that you can evaluate the form of reasoning separately from its content. He noticed that 'All A are B; all B are C; therefore all A are C' is reliable no matter what you substitute for A, B, and C. That let him build the first system for checking reasoning mechanically — the syllogism — and it produced the distinction logic still runs on: an argument is valid if the conclusion follows from the premises, and sound if it's valid and the premises are actually true. This is why 'All cats can fly; Socrates is a cat; therefore Socrates can fly' is perfectly valid and completely wrong. Aristotle had separated 'is this reasoning well-constructed' from 'is this reasoning about anything real,' and almost every reasoning error people make is a failure to notice which of those two things has gone wrong.