For many years India had only one major party – the Congress party. There have always been many parties, but most of them were either small parties or regional parties that had more influence at the state level than at the national level. In the 1970s, the communist and socialist parties emerged as a major opposition and leftist coalitions have formed the government occasionally. In the 1990s, the B.J.P. finally emerged as a second major party and is currently the head of the government coalition. There are two major coalitions now – the B.J.P.-headed one (which includes the educated and affluent middle class as well as Hindu fundamentalists), and the Congress-headed one (which includes secularists, minorities, and the poor). The leftist coalition, which represents intellectuals, socialists, and secularists, ranks a distant third.
The Congress party has always tended towards personality domination by members of the Nehru-Gandhi family, now represented by Sonia Gandhi, an Italian by birth, who is the widow of former prime minister Rajiv Gandhi (assassinated), who was the son of Indira Gandhi (also assassinated), and the grandson of Jawaharlal Nehru. I’ve been told that Sonia Gandhi is the most dangerous person in India. Her rivals in the Congress party have a habit of coming to an unfortunate end.
The B.J.P. is currently headed by the well-respected and moderate Atal Behari Vajpayee. He is quite old now and it is in doubt whether he can last another term. The B.J.P.'s problem is that it has no clear respected, centrist leader to take his place. There is a fear that without Vajpayee, the religious extremists in the coalition may gain ground.
You can get a good idea of party politics and governmental structures at this site.