A recurring element across anime, manga or works in other media. Tropes cover clichés, genre conventions and other recurring plot points, character types, narration styles, …
A trope, in the general sense used here, is a feature that is found in many creative works (anime, manga, or any other media), for example a plot point, a character type, a setting, a narration style, etc. It is expected that at least some readers or viewers will recognize the trope. Some tropes have a well-identified source that other works have imitated, departed from or alluded to; others are so old that their origin has long been lost.
The website TV Tropes collects tropes (in all media, including anime and manga, despite the name). If you're looking for a list of works using a certain trope, TV Tropes is the place to look. You'll also find lists of tropes seen in a particular work.