There are my personal definitions, stemming from my own observations.
-Thriller: Not entirely sure, but most stories that are labeled as "thrillers" seem to be action-oriented, only with horror elements.
-Horror: Any general exploration of the unusual and strange. The idea is that anything that is unknown can illicit terror (jump-scares not required). Personally, I would also add that such horrors serve to express the fragility of humans.
-Supernatural: A story with elements of what is considered outside the realms of our known sciences. They seem to refer mostly to existing mythos; demonology, angelology, cryptids, etc.
-Mystery: A story revolving around a question. The protagonist and friends would work to answer that question, and that makes the plot.
-Fantasy: A super-general, all-encompassing category, a fantasy is any story with major elements that go beyond realism. (Emphasis on major elements, because a lot of fictional stories are not fantasy because they don't rely on as much imagination.) When people think fantasy, they think LOTR or any old folklore with monsters. In reality, fantasy also includes almost all of Marvel's movies and Star Wars.
-Sci-fi: Stories that speculate the future of human civilization as technology progresses: Society, politics, culture, anything. Stories that are pure science fiction usually philosophize how humans will interact with technologies that will challenge our current "norms", such as Artificial Intelligence, alien encounters, genetic manipulation of humans, stuff like that.