In the 500s BC, during the later part of the Vedic period in India, the idea of reincarnation became very strong among Hindus. Most people believed that after you died, you would be reborn in another form, and then reborn again, and again, forever. But then people started to not like this idea. They didn't want reincarnation to just go on and on forever. Wasn't there any way to stop this; to get off the wheel of reincarnation and just be?
A young Indian prince named Siddhartha Gautama Buddha had an idea. He thought that you could get off the wheel of reincarnation if you were good and pure enough. He refused to be a prince anymore, and tried to spend his life being good and pure so he could get off the wheel. Gautama Buddha had many followers during his lifetime, and after he died he had even more. Most of Buddha's followers tried to be good while still living their normal lives - working in the fields or as soldiers, getting married, taking care of their parents and their children. But some of his followers adopted a Jain idea of getting away from the world so they could work full-time on being good and pure. These people were called monks or nuns.
In the 300s BC, one of the great Mauryan kings, Asoka, became a Buddhist, which helped Buddhism to succeed. Asoka convinced many other Indian people to become Buddhists. Buddhist monks started great monasteries, and some of these monasteries developed into centers of scholarship and research, like the great university at Nalanda in north-eastern India. By 200 BC, scholars were coming from as far away as China to study with the Buddhists at Nalanda. That's where, about 100 AD, scholars developed the numbers (1,2,3,4) we use today. But they don't seem to have heard about the research going on in Egypt, at the University of Alexandria, at the same time.
At first, most Buddhists were in India. But by 500 AD, under the Guptan Empire, travelling Buddhist monks spread Buddhist ideas west to the Sassanian Empire and east to China and other parts of East Asia. Travelling Buddhists also brought other Indian things like sugar with them to China. Chinese Buddhists kept on coming to India to study at the university at Nalanda.
By the 600s AD most of the Buddhists in India had gone back to being Hindus again. They still remembered Buddha, but as one of many Hindu gods. In West Asia, most of the Buddhists gradually converted to the new religion of Islam. The university at Nalanda began to be short of money and people, as Buddhism went out of fashion. Instead, scholars went to the newer Islamic university at Baghdad (in modern Iraq). By the 1200s AD, Nalanda had closed down.
In China, on the other hand, Buddhism got stronger and stronger. Soon most of the Buddhists were in China and not India. In China, as in India, most Buddhist people continued to lead more or less ordinary lives, but some Buddhist men and women left their jobs and their families in order to live in Buddhist monasteries as monks or nuns. But Chinese Buddhism was about meditation and action, not about scholarship and research. No new Buddhist universities opened there.