It is possible to enter Nepal by land via India or Tibet. Coming from Tibet, you must enter from Kodari and take a 7-8 hour bus ride from there to Kathmandu. There are several entry points from India. You can take a bus/train to any of the entry points on the Nepal border (Kakarbhitta, Birgunj, Dhangadhi, etc.) and then take another bus to your destination.
Tribhuvan International Airport in Kathmandu is the only international airport in Nepal. Flights enter Kathmandu from all parts of the world. See a brief list of the airlines below:
Thai Airways International operates daily flights between Bangkok and Kathmandu.
Qatar Airways flies daily between Doha and Kathmandu with connecting flights to/from Europe and America.
Gulf Air and Emirates have a daily flight between UAE and Kathmandu with connecting flights to/from Europe and America.
Air China and China Southern fly 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.