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 Wiki, "trope" has the even more general meaning of a recognizable pattern—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. 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 "tropable" 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.
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.
The Lives and Deaths of Tropes
Tropes are not eternal, sprung forth before the dawn of time from the brow of some literary god and like their forebear immortal and unchanging. Tropes are like many other things living and not in that they have a distinct life cycle.
All tropes began at some point as a clever idea that a Creator came up with in the course of creating a work. Other creators like the idea and use it in their works. Eventually it becomes a Trope -- a common, even standardized, component for the kind of work in which it appears.
Once it has been codified, a Trope may have an almost unlimited shelf life, appearing in works for centuries or even millennia. However, it may end up falling out of favor in some way -- overused, misused, prominently subverted, or even deconstructed so thoroughly it can't be reconstructed. Thus begins the twilight of a trope, in which it has become discredited.
When a trope has fallen so far that it is rarely used straight, and parodying or subverting it is a trope in and of itself, then it has become a Dead Horse Trope.
And if it is eventually left by the wayside, never used again except in period pieces, it has become a Forgotten Trope. And in this way a trope dies.