In general, an officer is a soldier who has authority over other soldiers. In modern militaries, a commissioned officer is an officer who is formally granted that authority by a government. This document granting such authority is called a commission, as in the officer is literally commissioned by the government to lead. A non commissioned officer is an officer who was not expressly commissioned, but has been promoted to the office.
What this means is that commissioned officers are, usually, trained at an academy, commissioned, then assigned a duty. These officers are generally trained for leadership or specific roles before they do them. Non commissioned officers enlist, are assigned a duty, then promoted to leadership as they prove competency/gain seniority.
Commissioned officers are the “educated” academy folks, the ones who are trained to lead, command, or perform specialized jobs such as fly a fighter plane, or command a platoon of tanks. Non commissioned officers are from the common soldiers, the enlisted troops. They start at the bottom, they do the “grunt work,” are the ones “on the ground,” and make up the bulk of the military.
In the modern military, they are paired together in a company to lead troops. For instance in the US army, each Captain has a First Sergeant that they are paired with to advise on their situations.
Commissioned Officers have very formal training, and generally have Bachelors degrees or higher in their area of specialty. But this generally makes them pretty young when they first are put in leadership positions. Where as enlisted soldiers don’t have to have a degree, (although most are now working on degrees at the same time) and they work their way to the top, so they have years of experience and training to help the Commissioned Officers make decisions.
This is with regards to a modern national military. Older militaries were very different in not just organization, but in their function and purpose.