One of the oldest storytelling devices in history, the climax is the point, usually in the third act (sometimes the second), in which the story reaches its very height, where the Conflicting forces collide in a manner that resolves the conflict — one way or another — with the Dénouement sweeping up the pieces. When the Big Bad and The Hero duke it out over the MacGuffin. When the detective has his showdown with the murderer. Etc.
Climaxes are often associated with a big fight of some sort (which frequently occurs in the action genre), but more low-key stories can have climaxes via The Reveal or other devices. It depends on the nature of the conflict that the story has been running on since the Inciting Incident.
In an arc-based TV or radio show or comic book, it is common to have a minor climax in each episode, and build up to a major climax at the end of the season or the series. Additionally, the final episode or two frequently make up a Grand Finale, a massive climax which wraps up all (or at least most) dangling plot threads.
In soap operas, each storyline tends to have a climax that brings together most of the characters involved, and is the dramatic high point of the arc. This climax often occurs in the middle of another arc; thus, when one climax is resolved, the show starts building up to another, separate one.
Often accompanied by Climactic Music.
The subversion of this trope is known as an Anticlimax.