The question, what are the responsibilities of a commissioner for corrections, does not specify any particular state, so the following is a general description. A commissioner for a state Department of Corrections is directly responsible for administering the state's system of prisons and correctional facilities for both adults and juveniles, although in most states the adult and juvenile detention systems are administered separately. As such, depending upon the state's population, he or she might be responsible for thousands of inmates as well as corrections officials and guards, administrative and support staff, and all contracting involving services for the various facilities. In addition, he or she may be responsible for oversight of the state's probation and parole systems, although, once again, many states have parole boards that administer that process separate from the administration of the corrections system.
Administering a prison and correctional facilities system involves more than simply housing and feeding inmates. It also involves prison industries, which often contract with governments to manufacture office furniture for government buildings -- which is cheaper than purchasing through commercial providers -- or license plates for vehicles registered in that state.
These are the responsibilities of the average commissioner for corrections in the United States. Obviously, this answer is specific to state prisons, and does not address the federal prison system.