There are three main Travel Modes of pet transport by air.
1) In-Cabin - Pet travels with owner inside the cabin. Weight limit is 8kgs of most airlines. Ideal only for cats and puppies.
2) Excess Baggage - Pet travels with the owner in the same flight as excess baggage, in the hold area of the aircraft. Weight limit 30-40 kgs for most airlines. Please check with airlines at source if they allow this option.
3) Cargo - Pet travels separately in another flight. Pet can enter India same day or only after owner has entered India. A registered agent at source may be required for booking with the airlines.
All modes are safe and widely used for pet transportation.