Enlisted

From Memory Alpha, the free Star Trek reference.

In Starfleet, an enlisted crewperson is one who has not completed the four-year Starfleet Academy course. In most cases, enlisted crewmen sign up directly to a ship or posting, and receive basic training, as well as any specific courses required for their position. (TNG: "The Drumhead"; DS9: "Shadowplay", "Starship Down")

Its predecessor, the Earth Starfleet, also had a hierarchy of enlisted positions.

Other force structures, such as the Klingon Defense Force and the Bajoran Militia, have enlisted men in their ranks, but it is unclear as to how they differ from their standard officer ranks.

As with commissioned officers, there is a rank structure amongst enlisted crew. In naval systems, these are referred to as "rates", in military systems these are called "grades". The term "non-commissioned officer" refers specifically to any enlisted personnel given authority over other personnel. In Starfleet, this could refer to any petty officer (up to master chief).

All commissioned officer ranks, from ensign upward, outrank all enlisted in the chain of command. However, in certain cases, noncommissioned officers can exercise authority over commissioned officers. (Miles O'Brien is one such case, in his position as chief of operations aboard Deep Space 9). Additionally, chief petty officers such as O'Brien wield influence far beyond their place in the rank structure due to their extensive experience and skill, and are expected to mentor junior officers as well as other noncommissioned officers.

Enlisted personnel may be promoted to receive a commission, but must pass necessary examinations and/or attend the Academy.

The Star Trek Encyclopedia had the TNG era enlisted insignia labeled as "warrant officers", but this seems to contradict several canon mentions.

Naval 

• master chief petty officer
• senior chief petty officer
• chief petty officer
• petty officers (specialists 1st, 2nd and 3rd class):
  • sensor analyst
  • technician
  • yeoman
• crewman (1st, 2nd and 3rd class)
  • The Star Trek Encyclopedia refers to these ranks with the non-canonical designation "ables'man". Some modern systems refer to crewman 2nd and 3rd class as "apprentice" and "recruit", respectively.

Military 

• first sergeant
• staff sergeant
• sergeant
• corporal
• private

Gene Roddenberry initially envisioned a Starfleet entirely composed of officers during the creation of TNG. As a result, there were negligible numbers of enlisted personnel seen in TNG. During DS9, the writers made a conscious decision to increase the number of enlisted personnel in Starfleet, to a level either equal to or exceeding that of the officers.

See also: List of Starfleet enlisted personnel