For Indian Nationals there is no permit required to visit Andaman & Nicobar Islands. Important to note here that visiting Tribal Reserve Areas in Andaman & Nicobar Islands is strictly prohibited. If someone is found doing the same one will be treated as criminal as per the law.
For foreign tourists Restricted Area Permit (RAP) is required to visit Andaman & Nicobar Islands. RAP for 30 days can easily be obtained on arrival at Port Blair from the Immigration authorities. RAP can be extended for 15 more days and that will be done by Superintendent of Police, FRO/CID, Port Blair.
You can find more information about entry formalities in Andaman & Nicobar official website. It is highly advisable to go through it before planning your trip.