Rules and regulations about visiting the Northeast States change from time to time. The following is the situation as of early 2019. Whatever permit you get, make several copies of it as you'll often have to hand one in at checkpoints, police stations and hotels.
Since 2011, Arunachal Pradesh is the only state for which foreigners need a permit. For Arunachal, the Protected Area Permit is valid for 30 days, costs the rupee equivalent of US$30 (US$60 for solo-traveller permits), and can be obtained in a couple of days through tour companies or, with a bit more effort, independently.
Foreigners visiting Manipur, Mizoram or Nagaland have to register with the police within 24 hours of arrival, but this is almost always taken care of automatically when you check in to a hotel.
Indians need an Inner Line Permit (ILP) to visit Arunachal Pradesh, Mizoram or Nagaland. These normally cost ₹100 or less and, depending on the state, can be applied for online, on arrival, or at the state's representatives in some other states.
Arunachal Pradesh You can apply online (www.arunachalilp.com) or in person at the Guwahati airport, Naharlagun train station or Arunachal government offices in Delhi, Kolkata, Shillong, Guwahati and elsewhere (www.arunachaltourism.com/inner.php).
Mizoram Available on arrival at the Aizawl airport or from some Mizoram government offices in other states (www.mizoram.gov.in/page/get-in).
Nagaland Apply online (www.nagaland.gov.in/portal/portal/StatePortal/OnlineService/IssueILPService) or in person at Nagaland government offices in other states (www.home.nagaland.gov.in/inner-line-permit-ilp).