If you want to enter in Nepal by land, you can get into from India and Tibet. From Tibet, you have to cross the border at Kodari and take a bus to Kathmandu. It is about 7-8 hours drive to reach Kathmandu, the capital city of Nepal. From India, take a train from Gorakhpur and from Sunauli/Belahiya border, then take a bus to your destination. To enter in Nepal, some other borders of India are Birjung/Rakshaul, Kakarvitta and others.
The only international airport in Nepal is the Tribhuwan International Airport in Kathmandu. There are several airlines that fly to Nepal. See the lists of those airlines:
Thai Airways International operates daily flight between Bangkok and Kathmandu.
Qatar Airways flies daily between Doha and Kathmandu with connecting flights to/from Europe and America.
Gulf Air has daily flight between UAE and Kathmandu with connecting flights to/from Europe and America.
Air China flies between Lhasa, with connecting flights to/from mainland China.
Austrian Airlines fly between Vienna and Kathmandu once a week.
Indian Airlines have frequent flights between Delhi, Calcutta and Kathmandu.
Turkish Airlines fly directly from Istanbul to Kathmandu.
Nepal Airlines fly -
between Hong Kong and Kathmandu.
between Bangkok and Kathmandu.
between Singapore and Kathmandu via Bangkok.
between Osaka and Kathmandu via Shanghai.
between Delhi and Kathmandu.
between Bombay and Kathmandu.
Beside these there are Biman Bangladesh, Jet Airways, Pakistan Airlines, connecting Kathmandu with many countries.