Other than Indian, Bangladeshis and Maldivian nationals, all visitors to Bhutan require a visa; all visas are issued from Thimphu; visas are only issued to tourists booked with a local licensed tour operator, directly or through a foreign travel agent. Applications for tourist visas are submitted by the tour operator. Visa clearance from Thimphu must be obtained before coming to Bhutan. Visa clearance takes at least 10 days to process. Air tickets to Bhutan cannot be purchased without visa clearance. At your point of entry the visa will be stamped in your passport on payment of US$20, two passport photos will also be required. Visas are issued for a 15day period; extensions can be obtained in Thimphu at a cost of Nu.510.
Indians do not require visa or any travel documents to enter up to Phuntsholing. Actually the buses from Siliguri go all the way up to Phuntsholing crossing the border gate. But beyond Phuntsholing, even Indians need a permit to visit other places within Bhutan. Indian citizens need to carry the passport (or any other government issued identity proof recognized by Bhutan Immigration) in order to get the permit. The permit is issued by the Bhutanese immigration office located in the border.
All other foreign nationals (except Bangladeshi and Maldivians) need proper visa to enter any part of Bhutan including Phuntsholing. Note that there is no 'visa on arrival' facility in Bhutan.