A commission is a document conferring authority to an individual. You might look at it as the President delegating a little bit of his authority to other persons to make decisions and get things done within clearly defined legal limits. An individual so endowed is a commissioned officer.

A non-commissioned officer doesn't hold a commission, but acts under the general direction of those who do. These are the corporals, sergeants and petty officers -- the ones who really do the work.

There is also a warrant officer. This person occupies a gray area between commissioned and non-commissioned officers. His authority is conferred by warrant, not commission, which delegates to him authority in a specific area -- such as army aviators, intelligence specialists, etc.

People in enlisted pay grades E-1, E-2 and E-3 have no specific authority -- these are the privates, airmen and seamen.