The office of the governor-general of India was established in 1773 when Warren Hastings was made the first governor-general of the presidency of Fort William, Calcutta, taking up the position in the following year.
Image Source: punjabjudiciary.gov.in/district/jalandhar/history/warren.jpg
Initially the office only had control over Fort William, but it quickly came to control the Bengal region of northeastern India.
The position was created because of the widespread belief that there was massive corruption in the East India Company that necessitated some form of British government oversight, which led to the Regulating Act. However, Warren Hastings, who held the position from October 20, 1774, to February 1, 1785, was himself subject to widespread allegations of corruption and was impeached in 1787, with the trial lasting from 1788 until his eventual acquittal in 1795.
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When Hastings left for England, Sir John MacPherson was made provisional governor-general until Earl Cornwallis (later Marquess Cornwallis) arrived in India to serve as governor-general from September 1786 until October 1793. He is perhaps best remembered for his surrender of the port of Yorktown in the American War of Independence. In India in 1792 he defeated Tipu Sultan at Mysore and was succeeded by Sir John Shore who retired in March 1798. His successor was the earl of Mornington (later Marquess Wellesley), who was the brother of Arthur Wellesley, the first duke of Wellington. Arthur Wellesley became the military adviser to the governor-general and established Fort William into a training college for the British administrators in India.
Marquess Wellesley left office on July 30, 1805, and was replaced by Marquess Cornwallis, who died two months after starting his second term. After a long period of Sir George Barlow serving as provisional governor-general, Lord Minto was appointed. The earl of Moira (later marquess of Hastings), who had served in the American War of Independence, was the seventh governor-general from 1813–23. During this time he oversaw the purchase of Singapore and also worked on improving the Mughal canal system.
However, he was forced out of office owing to a financial scandal, and his successor was Lord (Baron) Amherst (later Earl Amherst) who had led a British embassy to China and hoped to expand British possessions with his involvement in the First Anglo-Burmese War of 1823. Although the war was a British victory, large numbers of British soldiers were killed. In 1828 he was replaced by Lord William Bentinck, who had previously served in India as governor of Madras. His task in India was to massively reduce British government expenditure. He also tried to introduce some social reforms such as ending suttee, a tradition where widows sacrificed themselves on their husband’s funeral pyres. His successor, Lord (Baron) Auckland (later first earl of Auckland), quickly found himself involved in a disastrous war in Afghanistan and was replaced by Lord Ellenborough. When Ellenborough arrived in India, he received news of the massacre of the British in Kabul under policies introduced by his predecessor. Although his time as governor-general included a war in Sind, he himself was largely concerned with trying to prevent increasing Russian involvement in Central Asia. Sir Henry Hardinge was briefly governor-general from July 1844 until January 1848, presiding over British India during the First Sikh War.