There are several ways to define climax, since it has many applications in a variety of fields. In language and the humanities, climax generally refers to two separate things. It can be the moment of most tension, possibly most violence, or a solution that solves a narrative, or it can be the final argument in a series of arguments in a speech or essay. The word is Greek in origin, and means "ladder," with climax typically referring to the last step on the ladder. Everything beyond the climactic moment is a step down the ladder.
In narratives (short stories, films, plays, novels), a climax can be hard to miss. It shouldn’t be confused with the denouement, the series of scenes that may follow that moment of highest tension, plot resolution or great final action. You could look at most typical Agatha Christie mysteries to see the difference between climax and denouement.
In Miss Marple stories for instance, the climax is usually the moments just before, and right when Miss Marple uncovers the criminal. What follows afterward may be Miss Marple’s explanation of her thought process, usually to an audience of friends. Yet that moment when she may be in danger and the killer will be revealed is climactic in nature, and what follows can be strictly categorized as denouement, the summing up and explanation of why or how Miss Marple solved the crime.
Sometimes, a climax, instead of being a thrill and a moment of extreme tension, action or resolution, is framed as an anti-climax. The solution to a puzzle turns out to be very simple, or a character wakes up from horrific circumstances to find it has all been part of a dream. In the latter example, there may be climactic moments right before the dream ends, but the end can be so disappointing that it is considered anti-climactic. An anti-climax isn’t always a bad thing in a narrative, and sometimes it is intentional, but it can also be a clumsy way to end a narrative where the audience or reader feels cheated by the outcome. Imagine Jaws just swimming away instead of attacking the boat in the first Jaws film, or all the velociraptors running away instead of hunting down the survivors at the end of Jurassic Park.
Another way that climax is employed is in persuasive essays and speeches. There’s an old piece of advice that you should save your strongest and most persuasive argument as your last main point. This too is the last step on the ladder before you sum up, conclude or briefly outline what you’ve discussed in a conclusion.
Just as Miss Marple’s explanations are the denouement of each story, you could refer to your conclusion in an essay or speech in the same manner. Thus, when you’re trying to persuade, you want the final argument or point before the conclusion to really resonate. It’s your last chance as a writer or speaker to argue your point with the audience, so make it count.