After Alexander went back to Babylon in 324 BC, a man named Chandragupta was able to overthrow the old Vedic kingdom of Nanda and form a big new empire over all of northern India and into Afghanistan. Greek historians said that Chandragupta got the idea from Alexander. But it seems more likely that, like Alexander, the Indians got the idea of forming empires from the Persians, or just from earlier Vedic empire-builders in India. Chandragupta conquered the Indus valley back from the Greeks and as part of the peace treaty he married the daughter of Seleucus, who had succeeded Alexander, and gave Seleucus 200 elephants.
There was a lot going on in the Mauryan Empire. The Silk Road was getting started, and Indian traders were beginning to sell cotton cloth, pearls, pepper and cinnamon to their northern neighbors, the Sogdians and the Greeks, in exchange for fast horses and silver. These traders brought the new alphabet back to India, where people used their version of it to write down the Rig Veda, the Ramayana, and the Mahabharata for the first time. As India got richer from trade, the Mauryan kings built the first Buddhist stupas, and then carved Buddhist temples into the cliffs at Ajanta - because Buddhism was also growing rapidly under Mauryan rule. People in India began to use water wheels for irrigation, and they began to make steel to sell on the Silk Road. Even Hinduism changed, with apparently new ideas about karma and dharma coming into the faith, possibly from China. Old Hindu gods like Indra lost ground to new gods like Shiva and Vishnu.
Chandragupta died in 298 BC and was succeeded by his son Bindusara. Bindusara's son Ashoka made the Mauryan Empire even stronger, ruling some of southern India as well as the north.
But Ashoka's victories at Orissa were so bloody and awful, that (at least according to tradition) after that battle he gave up warfare for the rest of his life. Ashoka converted from traditional Indian Hinduism to the new faith of Buddhism, and he used his power to convince millions of other people to convert to Buddhism too, all over Central Asia.
After Ashoka died in 231 BC, though, his sons and grandsons were not as strong rulers as he was, and the Mauryan Empire gradually fell apart into a bunch of smaller kingdoms.
Learn by doing: check out a Buddhist temple
More about the post-Mauryan kingdoms
More about ancient India
Note: The first of these is a comic book, but it's a good story!