The first Europeans to establish roots in India since the fall of the Roman Empire were the Protuguese. Led by Vasco da Gama's landing at Calicut in 1498, they established themselves along the Malabar Coast, trading with the rest of the subcontinent from there. (The Portuguese maintained some holdings in India as late as 1961.)
In 1600, the British East India Company was given the right to a monopoly to trade with India. While the company's primary objective was to get spices from Indonesia (East Indies), they needed goods to trade for spices. The good they wanted was cotton, and they got it from India. In 1612, the English won a battle against the Portuguese. Because of this victory, they were able to gain the right to trade and establish factories in India from the Mughal Emporer. Because the Dutch controlled the East Indies, the English focused all their attention on India. The company traded for silk, sugar, and opium among other goods.
In 1664, French finance minister Jean-Baptiste Colbert launched the French equivalent of the English East India Trading Company. The French obtained a few cities such as Pondicherry and Chandernagore, and gradually expanded their trade. By 1740, however, this company's sales were only half those of the English East India Company.
The Danish, the Austrians, the Swedes, and the Prussians all tried unsuccessfully to get a piece of the action in India.