A commisioned officer is someone who has recieved training in leadership and management as well as combat (or what ever job they are employed to do by the army).For the UK (and probably the US and most other countries) this means that to be a commisioned officer you will need a university degree and will recieve additional training.
A non commisioned officer is someone who has been promoted due to experience and ability from the lower ranks ie a private gets promoted to corpral then to seargent.