Well, I didn't really know where to start with this one because of just how huge the East India Company was, but I thought I would give it a shot anyways, since it is something that I am currently studying.

At its height, the EIC operated not only as a trading company within the Empire, but probably most notably as a political and military organization in India. In India, in the year 1857 (as that is very relevant to the dissolution of the company) the EIC held the power to tax and govern the local population. They also had a large military in the Subcontinent, with the Bengal army being the most relevant in this case.

Over time the EIC succeeded in alienating the local population with heavy tax burdens, lots of overt racism, and insensitivity to local culture, particularly religious beliefs. What this culminated in was the Sepoy Mutiny of 1857, which began in the Meerut cantonment in May of that year. The Bengal army was made up of a disproportionate number of high caste Hindu troops, who had been offended by new practices which arose from ignorance or disregard of some practices. One of these practices was making these soldiers fight oversees, as seatravel for a Brahmin was seen as polluting and cause for loss of status. The catalyst for these tensions boiling over into open rebellion was the rumour of the use of pork and cattle grease in the cartridges of the new issue Enfield rifles. These cartridges had to be bitten open in order to allow for reloading. Consuming beef and pork was sacrilegious to Hindus and Muslims respectively, and was the cause for the Meerut incident which started the rebellion when 85 soldiers were to be punished for refusing to use the new rifles.

The rebellion lasted into 1858 and featured many incidents of reprisal on both sides against civilian populations. Cases of British women or Children being raped or killed were heavily reported on in Britain and acted to enrage the population. This led to heavy handed reprisals when British troops reinstated rule.

After the rebellion was over, there were opinions in the British government that the East India Company had failed, and with the "Act for the Better Government of India" passed in August 1858, the company was dissolved, and the crown took over most of their operations in India.

By 1874 the last vestiges of the company were formally dissolved, meaning that there are no companies that are to the East India Company as The Bay is to The Hudson's Bay Company. Since this was a complex issue, I was lucky that I happened to have a lot of sources already gathered on it, though if I missed any important information, or made any mistakes, feel free to correct me.

Bayly, C.A. Indian society and the making of the British Empire. New York: Cambridge University Press, 1988. 

Blunt, Alison. Embodying war: British women and domestic defilement in the Indian ‘Mutiny”, 1857-8. Journal of Historical Geography: University of Washington, 2000.

Brown, Judith M. Modern India: The Origins of an Asian democracy. Second ed. New York: Oxford University Press, 1994.

Streets, Heather. The Rebellion of 1857: Origins, Consequences, and Themes. Teaching South Asia, an Internet Journal of Pedagogy: South Dakota State University: 2001.

Levine, Philippa. The British Empire: Sunrise to Sunset. 2nd ed. Harlow, England: Pearson Longman, 2013. Print.

EDIT: Revision of concluding paragraph.

Could someone answer this question about the Dutch East India Company and not the British one? The Dutch one was in the TIL that I think this question might be referencing!