A Military Prison is a prison operated by the Military and is where people in the Military are usually sent for doing crimes against the military.
Even for just going AWOL you can be sent to the Brig or military prison.
Military Prisons are also used to house Prisoners of War, Unlawful Combatants, and even too house military personal that are deemed a security risk if they were to be allowed to roam free.
A Military Prison is usually much better and nicer than a state run prison that a local Judge sends you too.
A Regular prison is where anyone who commits crimes and is arrested by the local police would be sent to for things such as Robbery, Murder, Rape, Child Molestation etc.
Some people may be in Military prison for only a few months to a few years while they may also serve life in Military prison.