As the East India Company's power expanded from trade to rule and conquest, the British Parliament exercised increasing regulation and control over them, so it cannot be completely said, as some have done, that the East India Company was an autonomous army under the control of a company rather than a state. It was closest to this description from around the 1760s to the 1830s, as the huge territorial and military expansion of the EIC was gradually drawn under the power of Parliament. (I should note that, throughout this, "India" refers to British India, not the modern state of India, which is considerably smaller.)

Established in 1600, the Honorable East India Company was given a Royal Charter by Queen Elizabeth. This is important to note: throughout its history, the EIC operated under Royal Charter, which was needed for the company to have the legal right to trade in a defined part of the world. If Parliament were angered, it could refuse to renew the charter or even revoke it. In practice, this meant that part of the EIC's profits were paid to Parliament to keep renewing their charter. It is also worth noting that the stockholders of the company were chiefly English aristocrats (contrasted with the Dutch East India Company, whose stockholders were mainly merchants), many of whom were in the House of Lords, so just how separate the EIC ever was from the British state is already muddy.

At first, the company focused on trade, building "factories" (starting with Bengal and Java) that allowed them to gather trade goods, especially spices. At this point, they were competing with the French, Portuguese, and (especially) the Dutch for the lucrative spice trade. Treaties with Indian princes would give tremendously profitable monopolies to certain European powers, and soon this led to war. In 1612, the EIC defeated a small Portuguese fleet in the Battle of Suvali. All seagoing European ships at the time were armed with cannons to fend off pirates and competitors, so we can see why a trading company would also have naval power.

At the time, the preeminent power in India was the Mughal Empire, a Muslim empire claiming descent from the Timurids and Genghis Khan. With the patronage of the Mughals, the EIC was able to expand into several more cities, including Madras and Bombay (the English names for these cities, which are today Chennai and Mumbai). The EIC at the time was also clashing with the Dutch over the spice trade around what is today Indonesia. Through the 1600s and 1700s, there were several major wars between the Dutch and English East India Trading Companies.

It was Charles II, around 1670, who granted the EIC "the power to run mints, raise an army, make treaties, and exercise civil and criminal justice" (quoting from Sudipta Sen's Empire of Free Trade). Up to this point, the EIC's presence was mostly "factories" protected by forts. In the Child's War of 1686-1690, an attempt by the EIC to strongarm the Mughals into giving them more territory backfired, and the EIC was forced to surrender Bombay. After paying a huge indemnity and making a host of promises not to be so exploitive, the company was able to set up again in Bombay, as well as Calcutta (called Kolkata today). There was another incident in 1695, when the famous pirate Henry Every (or Avery) attacked the Mughal Emperor's fleet returning from Mecca. The British bungling of the situation increased Mughal beliefs in the English support of Every, and renewed hostilities ended with further reparations and promises. (Around 1680-1707 were a series of wars between the Mughal Empire and the rising Maratha Empire, in which the Mughals' failure to conquer the Marathas marked a shift in power in India.)

There was an attempt in 1697 to deregulate India, ending the Royal Charter's exclusive monopoly to the EIC. A second company was formed in 1698. Despite some competition between the two companies, it soon became clear that competition wasn't profitable for either, and they merged in 1708. This started a period of legal wrangling with Parliament, with Parliament wanting more control (and a share of the profits) and the EIC wanting more autonomy. In exchange for extensions on their charter, the EIC gave Parliament (and thus the English state) vast sums of money. In the 1720s, the Mughal Empire declined, and the Maratha Empire, among others, gained power. I mention this to show that India at the time was not unified, and the independent states warred with each other for dominance, often using the English and French as allies in their wars to gain advantages over each other, through which the European companies gained a number of concessions.

The Seven Years' War between 1756 and 1763 ended with France ceding control of its Indian interests to England. Under the leadership of the infamous Robert Clive, the Commander-in-Chief, the EIC expanded its control massively. The military strength of the EIC grew into the thousands, mostly native sepoy soldiers. Through military conquest and through absorbing the semi-autonomous "princely states" as "subordinate allies", the EIC gradually gained supremecy in India. They divided India into three "presidencies": Madras, Bombay, and Bengal. What was a force of around 5,000 in the 1760s expanded to 70,000 by 1796 and over 200,000 in the 1820s. It was over 350,000 in the 1850s.

In the 1770s, cheap Dutch tea imports in the American colonies left the EIC with massive debt and storehouses full of unsold tea. Parliament, called on to financially prop up the massive debts, passed a series of laws to make it clear that the EIC was under the control of the British government, starting with the Regulating Act of 1773 (later called the East India Company Act of 1773), Parliament made it clear that they had sovereignty over the EIC and all the territory it controlled. Parliament was also concerned by the invasions of Indian states and brutality of the EIC, including the famine in Bengal in the late 1760s. (Parliament also passed the Tea Act to force the American colonies to buy the EIC tea, and we know how that went over in Boston....) The East India Company Act of 1784 formalized this even more, creating a Board of Control in India under the Crown, and separating the EIC's political and economic functions. There were a series of more Acts that laid out the exact mechanisms of the balance of rule between the Crown and the EIC. Over the next few decades, Parliament stripped the EIC of its trading monopolies, and ultimately did away with its economic function entirely, leaving it essentially a territorial government. The Saint Helena Act of 1833 united the presidencies under the control of the Governor-General of Bengal, who became the Governor-General of India. As an interesting side-note demonstrating the increasingly regimented nature of rule in India, the East India Company College (today Haileybury and Imperial Service College) was founded in 1806 to train future members of the Civil Service, the civilian administration of India.

Richard Wellesley (older brother of Arthur Wellesley, who would become the Duke of Wellington), the Governor-General of India, invaded Mysore in 1799 and killed Tipu Sultan, the reigning Sultan of Mysore. This moment really shows why Parliament was so alarmed by the EIC, and why so many Acts of Parliament between 1773 and the 1850s were aimed at regulating EIC rule in India. It shows what amounted to a company director in India deciding to go to war more or less on his own initiative, to prevent Mysore from allying with France, then under Napoleon. After a series of three wars with the Maratha Empire from 1775 to 1818, the East India Company became the unrivalled power in India, with the exception of the region ruled by the Sikh Empire, which was conquered in 1849. (It's worth noting the Maratha Empire won the first war with the EIC in 1775-1782, but lost the second and third.)

Since this has become much longer than I intended, let me wrap up by the point this had to end up at: the Indian Rebellion of 1857. The subsequent Government of India Act of 1858 placed India under the direct authority of the British government.

Edit: Sources (in addition to the one I mentioned)! In case anyone wants to do more reading...

H. V. Bowen. 2006. The Business of Empire: The East India Company and Imperial Britain, 1756-1833
John Keay. 1991. The Honourable Company: A History of the English East India Company
Tirthankar Roy. 2012. The East India Company: The World's Most Powerful Corporation

There have been a lot of new books on the subject published in the last couple of years, but I haven't had the chance to get my hands on them. If anyone has read them and would recommend them, please let me know!