According to Wikipedia, Nepal has "open borders" with India underneath the 1950 Indo-Nepal Treaty of Peace and Friendship.
As far as I understand, it's not a truly open border as movement is restricted to Indian Citizens only.
By truly open border I mean, for instance, the Vatican City and Italy wherein anyone allowed entry into Italy is legally allowed into Vatican City even though the microstate is not a part of the Schengen Area or the EU.
Its open only for Indian or Nepalese citizens. Its not open for citizens of other countries.
This is a Nepalese link which says Indian citizen entering Nepal should have proper identity to prove that they are valid Indian citizens, that means no need of visa.
And this is a link from Indian Embassy in Nepal. On first line it says:
Introductory Information
Except Bhutan & Nepal Nationals all foreign nationals including children MUST obtain a visa to travel to India.
Hope this helps
And if you are travelling on a car or bikes or any means of transport from one country to another between India and Nepal, you have to make an entry at border check point but its always wise to get a letter(like No objection Letter for a car, bike) from respective embassy from current country.
Yes, an American, for example, need a visa to enter Nepal, which is not really what I would consider an "open border". Most countries allow some neighbors in without a visa (all the ASEAN countries, the US and Canada, the EU, etc.)
Yes Nepal has open borders with India, but it is only valid for Nepalese and Indian people. So it means citizen of either countries don't need passport and visa to enter each other's countries but they might need to show citizenship or other identity on ask.
India and Nepal have had free movement of people starting back in 1950. According to the treaty, there are are almost no restrictions on visiting, buying property, opening a company, etc. Foreign citizens aren't included in the scope of the agreement and the two countries remain fully separate in the eyes of most tourists.
The border between Vatican City and Italy is not 100% open - you can technically get a visa to visit the Vatican, but not Italy. Likewise Schengen countries sometimes issue Limited Territorial Validity visas, which are only accepted in one country. The border between the UK and Ireland is also not 100% open - each country issues it's own national visas, which aren't always accepted in the neighbouring country.
Similarly Canadians can only stay up to 180 days in the US, unless they extend their visa. Neither are Canadian/US tourist visas mutually recognized, nor is there a right to free movement between the two states. Nevertheless it's the longest undefended border in the world.
