When people in India began to follow the Buddhist religion, they needed new kinds of buildings that reflected Buddhist ideas. The earliest Buddhist building in India that we still have today is Sanchi, in central India. About 250 BC, the Mauryan emperor Asoka built a stupa there to keep some relics of the Buddha safe, the same way that later on King Louis IX built the Sainte Chapelle to keep relics of Jesus safe. The stupa was in the shape of a dome, but it was solid brick, like the earlier ziggurats of West Asia.
Probably people were also building Hindu temples out of wood during the Mauryan period, but because wood doesn’t last well, we don’t know much about these temples.
Beginning around 200 BC, people in western India, at Ajanta, began to cut Buddhist temples out of rock cliffs. Instead of cutting blocks of stone to build a temple, they carved the temple right into the rock, which was easier for getting the roof to stay up (compare this to the Chavin temple in Peru, a little earlier). These temples have a row of columns across the front, and a room in the back with statues of the Buddha in them.