Nepal being a landlocked country is surrounded by two Asian giants China in the North and India in the East, West and South. Hence travelling to Nepal via sea has come to end.
Secondly being mountainous country most of its land is covered by steep hills and mountains. Hence train transport is difficult to construct and it is limited to few part of Terai region which is also limited to freight transportation.
Therefore land and air transport are the only remaining means of transport between Nepal and India.
There are several airlines like Air India, Nepal Airlines, Himalayan Airlines, Jet Airways, Vistara Airlines, Indigo Airlines etc which has regular daily flights to Kathmandu from various parts of India. Airfare ranges from almost USD 150 for an round trip.
Another popular but little adventurous way to enter Nepal is by land transport. You may prefer Train + Land transport or direct bus service from various cities in India to Nepal. Normally it takes 24 hours to 35 hours to travel to Nepal from various popular Indian cities.