There was never one war that gave the British control of India. The English first arrived in India to stay in the early 1600's, in the form of the English East India Company. In 1640 it acquired the city of Madras, and in 1661 the city of Bombay was transferred from Portuguese to English hands as the part of a dowry and then turned over to the East India Company seven years later. The Company would fight limited wars with the Mughal and the Maratha Empires in this period as it established itself in the Indian economy, and started to make tremendous fortunes. In this, local elites cooperated with the English in making them both very rich indeed.

British and French rivalry in Southern India would begin the next stage of European interventionism. When the French took Madras from the British their ally, the Nawab of the Carnatic, intervened. His army of ten thousand cavalry was unable to defeat a French force of 500 men in two separate engagements and demonstrated the superiority of European infantry to the dominant military arm of India. The British and the French would then fight a proxy war over the rule of two nearby nawabships, both installing their own nawab in one of the two contested areas, ruling them by proxy. The next great step was the conquest of Bengal by Robert Clive after the Nawab of Bengal stormed Calcutta. In seven months Clive took the richest province of the Mughal Empire for the East India Company, installing a puppet nawab to rule Bengal. Nine years later the British would take over direct control of Bengal after the Company was inducted into the Mughal hierarchy and made the chancellor of Bengal.

The British conquest of India would continue through various wars into the 19th century, but also through the establishment of treaties and client kingdoms. While the Maratha Empire would fall in the Third Maratha War, the Mughal Empire would remain until the 1850's when it would be involved in the Great Indian Mutiny and as a result dissolved. The final wars of the conquest period would be fought with the Sikhs and Muslims of modern-day Pakistan in the 1840's. In all these wars the East India Company was helped by the political fragmentation of the Indian subcontinent, and the willingness of Indians to accept Company rule and to work with them. While there had existed large empires, they were riven by internal dissensions that the Britsih exploited, and were unable for the most part to present a unified front to the Companies armies. Many elite and common Indians worked with the British, benefiting from the greater stability found in British Indian than outside of it. The armies of the East India Company were mostly made up of Indians drilled in the European style and led by British officers. It was these Indians that conquered India for Britain.

The British were successful because they became a part of Indian society so effectively. Local rulers obeyed the British in much the same way their ancestors had obeyed Mughal Emperors. The greatest challenge to British rule only came in 1857 with the Great Indian Mutiny. The causes of the Mutiny are still debated, but many tie the support the mutiny found among local elites to the changes in British policy that were a direct attack on their interests. Under Governer-General Dalhousie the East India Company annexed several of their client states after refusing them the right to adopt heirs, raising concerns among the Indian nobility about their future under British rule. In addition, he laid duties on giving landlords title and confiscated thousands of estates. British rule in India would survive the mutiny, but the East India Company would not, as the British government took over the government of India, created the Indian Empire, and established Queen Victoria as Empress of India.

Throughout this period there had never been any attempt at colonization such as in North America. The British found the Indian climate decidedly unhealthy, so there were never very many in the subcontinent at once. Occasionally East India Company officials would bring their wives and children with them, but most were unmarried and had arrived to make their fortune before returning to Britain rich men. Many took local mistresses, which helped integrate them into local society. Only with the advent of steamships and the creating of the Suez channel did many European women travel to India, and few came to think of it as home. Most were there only to serve as administrators and then returned when finished, and they sent their children back to Britain to be educated, so these children would not have seen India as home either.

For further reading, I suggest the books I used as sources for this post

John Keay, India: A History : From the Earliest Civilisations to the Boom of the Twenty-First Century, 2010, for a modern overview

and

Edward John Thompson and Geoffrey Theodore Garratt, Rise and Fulfilment of British Rule in India (New York: AMS Press, 1971) for a more contemporary overview. The 1935 edition is available on archive.org at https://archive.org/details/in.ernet.dli.2015.210780

So, the main turning point for Indian conquest was by the East India Company (EIC) in Bengal at the Battle of Plessey in 1757. Background for this battle: The EIC already had a port and stronghold in Calcutta, Bengal at the time. The Nawab of Bengal, Siraj-ud-daulah, was attempting to recapture the port city due to the fear of EIC but he was also a bit overzealous. He forced the main banker family of the area, Jagat Seth, to fund his campaign. The Nawab was successful in recapturing Calcutta but there was a major issue in dealing with the British civilians at the time. So they were sent to a holding cell. However, this cell was not spacious enough to hold all of the prisoners. This event was known as the "Black Hole of Calcutta" and the news of this event spread back to the EIC and Britain. This forced the British to send reinforcements with General Robert Clive. Now, we get to the Battle of Plessey. So, the Jagat Seth, tired of funding the army of the Nawab, conspired with Mir Jaffar, the general under the Nawab, and Robert Clive. Siraj-ud-daulah was defeated and Mir Jaffer was made the Nawab of Bengal. This made this position a puppet figure and the region was theoretically under the control of the EIC. This led to the EIC conquering the rest of the Indian subcontinent via force or treaties with the princes of the regions. (Also, a main point to note is that the Mughal empire had little to no power over the whole region, and was a puppet figure). There were some other hold-outs in the other regions such as Mysore but the EIC won out in the end. However, this, at this point, the EIC has power over the region in the guise of the Mughal Emperor's protection, not the British monarch directly. The EIC did receive orders from the empire but they did have some level of autonomy. This changed after 1858 by Queen's Victoria's Proclamation which officially brought the Indian subcontinent under the Crown's control. If you want more info on why this happened, I can explain that more in the depth.