The decline of the Mughal Empire caused a vacuum into which the Europeans moved. The first Europeans to reach India by sea were the Portuguese who arrived in 1498 and began importing spices from India. They formed a base at Goa in 1510. However in the 17th century the Portuguese declined and the English and Dutch took their place.
The English East India Company was formed in 1600 to trade with India. In 1639 the English established a trading base in India. Itgrew into Madras. In 1662 the English king married a Portuguese princess and he was given Bombay. In 1668 it was sold to the East India Company. In 1690 the English established a base in Bengal, which grew into Calcutta. In the late 17th century the Dutch also declined and the French replaced them. In 1673 the French established a base at Pondicherry.
In the 18th century French and English became bitter rivals and they both began to interfere in Indian politics.
The Seven Years war between Britain and France began in 1756. With the outbreak of war the Nawab (ruler) of Bengal, Siraj-ud-Daula captured the British base at Calcutta. Notoriously he forced captives into a small cell and most of them suffocated overnight. This became known as the Black Hole of Calcutta. The East India Company sent a force led by Robert Clive (1725-1774) to recapture Calcutta. They soon did so.
However Clive was not satisfied and he decided to take the whole of Bengal. Clive won a great victory at Plassey in June 1757. (The battle was won largely because one of the commanders of the Bengali army, Mir Jafar, changed sides and refused to join the battle). Clive then overthrew the ruler of Bengal, Siraj-ud-Daula and replaced him with Mir Jafar. However Mir Jafar was only a puppet.
In 1765 the company began to rule Bengal directly. Clive's victory at Plassey ensured that India would eventually become a British colony not a French one. However the Company did not take over India straight away. It was a gradual process, which took several decades. The East India Company eliminated French influence in India and began to subdue other Indian states.