There were a number of European powers that tried to exert control over India. However, the only one that can really be said to have succeeded was the United Kingdom. The British essentially ruled most of India (along with what is now Pakistan and Bangladesh) for almost two hundred years. The other European powers that tried to have control in India were the Dutch, the Portuguese, and most especially the French. The British had to fight a series of wars against the French before they ousted the French from India.
All of these powers were interested in India largely for economic reasons. At first, the attraction of India was the spices it grew (especially pepper) and its proximity to areas like Indonesia where other spices were grown. This was a very big business for the Europeans for centuries and led them to try to get footholds in India. India was also (of course) a very big country with a variety of resources. The Europeans wanted to have a monopoly over those resources. Finally, they wanted to have captive markets in which only they could sell goods that they produced in their home economies. India’s huge population was very tempting in this way.
The powers also wanted India simply for prestige. The Europeans were very interested in having large empires that would prove the strength of their countries. Control of India was a good way to do this.
Overall, though, economic factors were the main driving force behind European attempts to control India. The British were the ones who were most able to exert that control