Nobody knows what Indian government was like in the Harappan period. Probably they had a king over each city, the way West Asia did at this time. Probably they had some queens, the way Egypt did. But about 1500 BC, there was an Indo-European invasion. Sometime after that, the government seems to have divided up all of the people in India into castes. We know this from the Rig Veda. Only men from the highest caste – the Brahmins – could be rulers in the government, or even ministers or clerks in the government. These men did not let women rule either. Most of India was still divided into small kingdoms. Each kingdom still had its own king.
Starting in the 300s BC, some of these kings formed larger empires in India, especially in the north. During the 300s BC, Chandragupta conquered a lot of northern India and made it into one big empire. Chandragupta’s Mauryan Empire lasted until about 200 BC. But then the small kingdoms of India regained their independence. The same thing happened again in 319 AD, when Chandragupta II formed the Guptan Empire. And then in 455 the Guptan Empire also collapsed. After that, the most important kingdom in India was the Chola kingdom. The Chola ruled all of south India and east India along the coast. Then in the 1100s AD the Abbasids invaded and made northern India part of the Islamic Empire, under the Delhi Sultanate. Meanwhile the Pandyas took over southern India. So during the medieval period, India went back and forth between being ruled by small kings and being ruled by larger empires.
Within the Mauryan and Guptan empires, and in the Chola kingdom, the kings appointed a lot of administrators to help them govern. The empires were divided into many smaller provinces. Each province had a governor to rule it. That’s the same as in the earlier Persian Empire or in Ch’in and Han Dynasty China about the same time. Within each province, the governors had councils of ministers to advise them. The governors sent out tax collectors to collect an income tax and custom duties. (That’s a tax on anything you brought into the kingdom to sell.) Other officials kept track of birth and death records, and the government budget. But all of these governors were controlled by the king. And the king kept a standing army of thousands of soldiers in case any of his governors rebelled.