This is an essay.
It contains the advice or opinions of one or more Wikipedia contributors. This page is not an encyclopedia article, nor is it one of Wikipedia's policies or guidelines, as it has not been thoroughly vetted by the community. Some essays represent widespread norms; others only represent minority viewpoints.
There are a lot of decisions made on Wikipedia, as the encyclopedia constantly changes and grows. Every decision that is made by every editor on every article might be disputed by another editor. This often leads to an argument. Thus, Wikipedia plays host to very many arguments.
Anatomy of an argument
An argument may involve any number of editors, but most of them, at least when they start out, involve two people: You, and someone – let's call them Editor Q – who disagrees with you about some edit.
The argument may or may not start with a reversion. This is normal, per the Bold-Revert-Discuss cycle. After whatever edits are made and reverted, you find yourself on the talk page, having arrived at the "D" of "BRD". This is where the magic happens!
It begins in a straightforward manner: either you or Editor Q makes a case for their edit, and there is some response from the other, making a case for their edit. At this time, one party may be convinced by the others' reasons, and then the matter is settled. Understanding has been reached, and editing can continue peacefully. More often, neither participant is convinced by the other. In this case, we have finally arrived at the argument proper.
Lots of things can happen at this point. You and Editor Q might advance further claims in support of their position. Policy citations are common, as are citations of primary or secondary sources. Either participant may attack perceived weaknesses in the other party's argument. Once these things are happening, the argument is in full swing, and at this point, it becomes less likely that one party will convince the other.
Personal investment – Nobody wants to "lose"
The reason for this is simple: The effort that you've put into defending your edit, or attacking Editor Q's edit, represents a personal investment in the argument. If you don't "win" then you'll "lose". Not only that, but you'll "lose" to Editor Q, who seems less and less charming with each round of argument.
This effect, of growing attachment to your position, can become heightened in cases where the argument has strayed from an academic discussion of edits, and gone into personal territory. Once Editor Q has called you a troll, attacked your position as bigoted, and threatened to "take you to ANI", you really don't like the guy, and you really aren't going to let him win! This is where arguments get increasingly frustrating, and it becomes easier and easier for them to go right off the tracks, and crash amid blocks, page protections, and bitter recriminations.
You don't have to "beat" the other guy
There is good news. Even though you are very unlikely to convince Editor Q of anything, you don't have to. Before the argument gets to the point where you want to beat him over the head, as well as beating his arguments, remember that Wikipedia is a very big place. There are lots of people here, and for any particular topic, most of us are more-or-less neutral and open to learning things. Even though you can't extract an admission of defeat from Editor Q, if six other editors show up and take your side, then your edit is going to stick, and you won't even have to revert Editor Q. Someone else will do it for you.
Of course, if a consensus of uninvolved editors comes out in support of Q's edit, then you'll reconsider your edit. If you're still of the same mind, and you can't persuade anyone, then you can either continue to work to build consensus for your position, or else decide to move on.
It's possible, of course, that outside opinions will not line up all in agreement with you, or all in agreement with Editor Q. They may be divided between the two positions, or someone may introduce some third option, or some compromise edit. In such cases, the issue is more complicated, and it is especially valuable to have input from many editors, to help feel out the possibilities.
Cooling the situation down
In general, think of a metaphor involving heat: if you have a quantity of hot gas, what's the fastest way to cool it? Recalling the ideal gas law, we allow the hot gas to occupy a larger volume, and this causes the temperature to drop. Discussions are the same. In this metaphor, the gas represents the ideas, and the number of heads represent the volume. What might become an all-out-fight between two editors is likely to be handled more coolly by twenty-two editors.
Where do you find these people?
When applying this strategy, you have to be careful about how you bring new people to the situation. We discourage canvassing for votes or for support, in the sense of "rallying the troops", or bringing supporters of some edit in great numbers to overwhelm the opposition. You don't want to do this, so there are good, neutral ways of bringing more parties to a dispute.
Your first option, in the case of an argument between you and one other editor, is Wikipedia:Third opinion (WP:3O). When you request a third opinion, remember to state the dispute as neutrally as you can. Note also that it's considered polite when requesting an opinion, to give a third opinion on some other listed dispute.
After 3O, the next option is to request comments from the community at large, or from specific specialized parts of the community. The generic option is Wikipedia:Requests for comment; for a more specialized audience, consider finding a related WikiProject and posting a request for input at the talk page there. If there is a particular policy or guideline that is in dispute, the talk page of that policy or guideline could be a good place to look. If you're still unsure where to ask, try posting a general query to the Wikipedia:Village pump, and see what kind of reply you get there.