Essentially a tropical destination, India weather is typically hot and humid. However, the country is so large and diverse that the stunning cold of the Himalayas can coexist with the searing heat of the Rajasthan Desert.
It's best to think of the climate in India in terms of hot, wet and cool. The amazing heat, which can reach 125 degrees in areas of Central and Western India begins to build in April. In June the monsoon rains bring some relief, but downpours can make for calamitous travel conditions and it's still very hot and humid. Then in October the rains cease and India is at its cool and lovely best.
In general, the north tends to be slightly cooler than the south and central plains. Naturally, any place with a high elevation, such as the hill stations of Darjeeling, will enjoy a cooler climate.
In short, November through February are the best months for travel as the weather in India is optimal, though both October and March can be pleasant as well. If you plan to travel to the Himalayas, shoot for September or October.