A shorthand was a simplified term or group of phrases used amongst people in the same profession. It could also develop between individuals who had a shared history.
Trip Tucker and Jonathan Archer developed a shorthand after spending a lot of time together. A hologram of Travis Mayweather told William T. Riker while Riker was playing Enterprise NX-01's Chef in 2370. (ENT: "These Are the Voyages...")
In 2374, shortly after becoming acquainted with Jack, Patrick, and Lauren, other Humans who, like himself, had been enhanced neurologically, Julian Bashir found himself connecting with them, speaking in shorthand and finishing each other's sentences. It was an experience such as he'd never had before with anybody else he knew. (DS9: "Statistical Probabilities")