How to get entry permit for protected areas in India?

Originally your question was about wildlife protected areas. Such areas in India are designated as Wildlife sanctuaries or National parks. These areas, while out of bound for habitation or economic activity, are very much open to foreign and Indian visitors to explore and watch wild animals in their natural surroundings. The entry rules for these parks are drafted by respective managements and while they may have restrictions on the number of people allowed or the kind of equipment you can bring, you can just walk to the ticket counter and buy a ticket (although for foreigners prices are generally 6-10 times higher than for Indians).

• If you wish to watch Asiatic lions at Gir National Park, you can buy a "permit" for a safari online.
• Kanha National park also only stipulates that foreign visitors have to pay double the fee.

If you wish to visit wildlife sanctuaries or national parks, I suggest booking a resort which is near such a place and then the resort managemet will be able to provide you with information about any additional documentation/restrictions which may be in place. But. generally they are open for all.

The other protected areas

Certain areas near India’s external borders have been notified as Restricted or Protected Areas (The edited link in your question) for security reasons. Foreign national and, in some areas, Indians who are not residents of such areas need a special permit from the Home Ministry (Which deals with Internal Security in India) to visit such areas.

Depending on whether the area is a Restricted Area or Protected Area, you need a Restricted Area Permit (RAP) or Protected Area Permit (PAP).

Why would someone want to visit such areas? Because these areas have gems such as

• Leh
• Spiti Valley
• Havelock Islands
• Nathu La etc.

This FAQ issued by Ministry of Home Affairs answers most of questions about RAP and PAP. Quote:

Under the Foreigners (Protected Areas) Order, 1958, all areas falling between the ‘Inner line’, as defined in the said order, and the International Border of the State have been declared as a Protected Area. Currently, Protected Areas are located in he following States: –

(i) Whole of Arunachal Pradesh
(ii) Parts of Himachal Pradesh
(iii) Parts of Jammu & Kashmir
iv) Whole of Manipur
(v) Whole of Mizoram
(vi) Whole of Nagaland
(vii) Parts of Rajasthan
(viii) Whole of Sikkim (partly in Protected Area and partly in Restricted Area)
(ix) Parts of Uttarakhand

MHA has delegated powers to various authorities to issue requisite permits to foreign visitors. The authorities who can grant such permits varies for each region. You can get in touch with local Indian mission to find if they have the power to issue such permits for the areas you wish to travel. A full list of authorities for each place is on immihelp. For some states such information is also provided on respective tourism websites:

• For Sikkim, you may obtain the permit from Indian missions abroad.
• For Andaman & Nicobar Islands too you can obtain the permit from abroad or on arrival by air at Port Blair.
• For Leh and Ladakh permits are issued only by District authorities at Leh and the conditions are more stringent.

Generally applications for PAP must be made in groups(2-4), with an authorized guide and pre-defined itinerary (Source from Leh PDF. However, if you are a citizen of China, Pakistan, Afghanistan, or are of such origin, or a solo traveller, your case will be referred to MHA and may take significantly longer. Source: From MHA FAQs

In all cases relating to grant of PAP/RAP for an individual
foreign tourist, the proposal would require prior clearance from the
ministry of Home Affairs. The requisite clearance would be conveyed
by MHA only on the basis of the recommendation of the State Government
concerned.

AND

With a view to promote tourism, some areas (notified by the
Government of India from time to time) can be visited by
foreign tourists, either in groups, or as a couple in the case of a
husband and wife, or by individuals, after obtaining the necessary
permit from the competent authority. Necessary powers have been
delegated to various authorities to issue the special permit,
without the prior approval of the Ministry of Home Affairs,
to facilitate foreign tourists subject to the following
exceptions:

(i) Special instructions applicable to foreign diplomats and members of the United Nations and International Organisations holding Diplomatic / Official Passport are issued by the Ministry of External Affairs.

(ii) Citizens of Afghanistan, China and Pakistan and foreign nationals of Pakistani origin, shall not be issued a special permit without the prior approval of the Ministry of Home Affairs.