A noncommissioned officer (henceforth "NCO"), called or translated as "sub-officers" in some armed forces of the world, are the backbone of any organized military force, and NCO-alikes are found in almost every armed force, regular or irregular, throughout human history.
NCOs are typically senior enlisted personnel, inferior in the chain of command to all commissioned officers, but holding specific legal authority above and beyond that of regular enlisted personnel, often including the authority to dispense and authorize nonjudicial punishment and administrative action.
In the coarsest of terms, and in the general case, NCOs are responsible for enforcing and directing the completion of orders given by commissioned officers. On a daily basis, this generally means they act as supervisors, trainers, and bureaucrats. Commissioned officers very often fill these same roles, but the major difference - the commission itself - is an important legal distinction and is in fact set forth in both public and military law in the United States as well as many other countries.
In the United States armed forces, a military officer receives his commission by legal authority of the President, who is Commander in Chief of the armed forces. Officers are therefore empowered to give legal orders, which must be followed by subordinates, whereas NCOs may not give orders - they may only enforce them.
For example, a Captain in the United States Army - a junior commissioned officer - may order that the barracks be cleaned. He would likely not wish to order thirty individual troops to perform individual tasks. It would then, in this example, fall onto an NCO to execute that order. This is called a delegation of authority.
The NCO would be in charge of organizing and directing troops in specific tasks, perhaps by giving specific instructions to individuals. He or she might tell a group of soldiers, "You go clean the latrine," and another "You sweep and mop," and another "You buff the floors", etcetera.
He or she could also possibly delegate authority further - a Staff Sergeant may put one Corporal in charge of the latrines, another in charge of the sleeping bays, and another in charge of the equipment room. Those Corporals would then gather other lower enlisted and direct them in specific tasks, ensuring completion of the sub-tasks before reporting back to the Staff Sergeant, who would in turn report to the originating Captain.
The example of barracks cleaning is, obviously, very simple - the web of authority in actual organizations is very often dictated by the commanders of the organization, and supersets of specific policies. In theory, any commissioned officer can give orders to any NCO or lower enlisted person - in practice, it is seldom the case that an officer would issue direct orders outside his or her chain of command.
The same is true of non-commissioned officers, and their involvement with troops outside their chain of command. As their legal authority is in most cases derived from that of commissioned officers in their chain of command, they typically have little so-called "real authority", however the military tradition of most forces vests them with the power and responsibility to police order and discipline anywhere in the ranks, regardless of unit or command.