Intuition says that, whether you want a blazing row or a genteel debate, arguing is an easy thing to do. Intuition is wrong. Many, many people can't argue, and for a keen arguer, it's the most frustrating thing in the world trying to debate something with them.
Oh, anyone can shout their opinion from the rooftops, and anyone else can shout a different opinion back. That's not an argument, that's a shouting contest. The person with the loudest voice, or who gets fed up last, wins.
On a slightly more sophisticated level, both (all) parties attempt to give justification for their views. This is what you might call a proto-argument: getting there, got the basic principles, but it doesn't really work. The thing is,
the other person also has justification for their view.
You see, the aim of an argument is not merely to make your opponent stop talking, but to convince them. Once in a blue moon, possibly, someone changes their mind simply from hearing the justification for a view they don't hold.
To really get anywhere, to have a good, satisfying, fulfilling argument, you need to address not merely your own argument, but theirs. By giving your own views and reasoning, you at best show that there are two different opinions to choose from. They're not just being obstinate, it's mathematically unsound to change your beliefs without evidence against them. To change a mind, you have to show not only that your view is valid, but also
that theirs isn't.
You need to find the hole in their argument. If there isn't one, they may be right.