Gonzo journalism is a style that does not claim objectivity, it primarily just tells the story the reporter sees as they see it, with the reporter possibly part of the story, possibly even the protagonist.
Often energetic, first-person, in the moment. Also frequently associated with social critique and self-satire.
The impression most readers have is framed by contrast with classical journalism - more interesting, often less objective in that it includes the author's opinion, arguably sometimes more objective, or at least unedited, verbatim, raw, rather than aligning to a specific publication -- Hunter S Thompson, who coined the term, argued that in particular politics is too often reported with a pretense objectivity, where certain claims will never make it to print for narrative reasons (in any sense of the word, from readability to politics, to things that can be better understood by immersion than by reading statistics about it), even if it's things that writer and reader are both thinking.
It later developed into a telling-it-like-it-is style of journalism, somewhat more refined than Thompson's.
Gonzo other things
It's mostly the above senses that carries into other areas.
- gonzo politics or politicians - those who are rough and to their own point, without any pretense at balanced views
- gonzo pornography - no prettification, just put the camera into the action, possibly by the actors themselves. No separation between camera as in cinema, in terms of storytelling or typically physical distance, and probaably little editing,
- note that in more mainstream media, gonzo is often pretense at such, because it sells. Consider so-called reality TV.