A non-commissioned officer (also known as NCO or noncom) was an enlisted member of Starfleet who functioned as a lower-ranking crewman with authority over other enlisted crewmen, sometimes referred to as a team leader, and may serve as a member of starship or space station's senior staff. This rank was the opposite of a commissioned officer.
Sirna Kolrami dismissed Commander William T. Riker's choice of having Acting Ensign Wesley Crusher serve aboard the USS Hathaway during Operation Lovely Angel, underestimating the youth, stating, "I saw no sense in Riker choosing him anyway, he is just a non-commissioned child." (TNG: "Peak Performance")
In 2372, the Jem'Hadar first Goran'Agar correctly identified Chief Petty Officer Miles O'Brien to "what Starfleet refers to as a "noncom."" As such, he identified O'Brien as "a priority target" because he had "a great deal of experience." (DS9: "Hippocratic Oath")
As Starfleet ranks in Star Trek mostly based on the ranking systems of "real world" militaries, the following may be true: petty officer (including specialists, yeomen, and technicians) would fall under the category of NCO, as they were senior to the lower crewman rank. This included the more senior NCOs, such as chiefs.