You've seen this somewhere before…
Merriam-Webster gives a definition of "trope" as a "figure of speech." In storytelling, a trope is just that — a conceptual figure of speech, a storytelling shorthand for a concept that the audience will recognize and understand instantly.
Above all, a trope is a convention. It can be a plot trick, a setup, a narrative structure, a character type, a linguistic idiom... you know it when you see it. Tropes are not inherently disruptive to a story; however, when the trope itself becomes intrusive, distracting the viewer rather than serving as shorthand, it has become a cliché.
On This Very Wiki, "trope" has the even more general meaning of a pattern in storytelling, not only within the media works themselves, but also in related aspects such as the behind-the-scenes aspects of creation, the technical features of a medium, and the fan experience. The idea being that storytelling is not just writing, it is the whole process of creating and telling/showing a story.
Etymologically, "trope" is from Greek, meaning "turn".note
Around here, it is a stunt root, as in, "That isn't really different enough from our other tropes to be separately tropeable." Whether or not a subject is a trope is referred to as being "tropeable" or "tropeworthy"; works that are particularly tropeable are often referred to as Troperiffic.
The intent being to set Noah Webster spinning in his grave as quickly as possible.note
Don't let all this give you the impression that we exactly invented our sense of "trope": the more or less synonymous expression "resonating tropes" long pre-existed the site and community here, and you will find people outside of and independent of the site using the word "trope" in the same fashion that we do. Note that currently the Oxford English Dictionary actually recognizes the definition "a significant or recurrent theme; a motif", its earliest quotation for this meaning being from 1975. Merriam-Webster also somewhat recognizes this meaning, but twists it into "a common or overused theme or device: cliché", which seems unjustly condemning. Wiktionary, on the other hand, shows that our use of the word was used in 1776, meaning the word "trope" is the same age as America.
We also didn't invent the notion of finding and cataloging as many tropes as humanly possible or the idea of all media being formed around the same set of core tropes. A perusalnote of our Books on Trope page and the works linked there will show you that people have been identifying and discussing patterns in media for centuries. The first troper on record was, in fact, Aristotle. Yes, THAT Aristotle.
See Playing with a Trope for a comparison of the ways that a trope can be used.