MST stands for 'Mystery Science Theater', but has more or less become a word in its own right. It refers to Mystery Science Theater 3000, or MST3K, a popular television programme. To MST something is to give it the MST3K treatment – that is, to spork it by injecting commentary with the intent of being funny.
In the show, a guy called Joel (and later, a guy called Mike) was trapped in a satellite with a bunch of talking (and sarcastic) robots – namely, Tom Servo, a small but thick red torch with a gumball head and a soothing baritone, and Crow, a crazy, semi-cosplaying robot with funny eyes and a bowling pin beak. His bosses back on Earth, the 'Mads' (both Dr. and Pearl Forrester) and their henchmen, sent him cheesy B-movies and monitored him while he watched them. Joel/Mike and the 'Bots made sarcastic, usually humorous comments about the movie while they watched (sometimes referred to as 'riffing') as an attempt to save themselves from complete madness – though said madness has claimed Crow at least once.
In fanfiction, an MST is what you get when an author goes through a piece of fanfic and adds their own comments between the lines, as either themselves or other characters. This is usually merciless. Originally, MSTing used the characters from the show, but since then people have branched out.