The airport in Mumbai (Bombay) is called Chhatrapati Shivaji International Airport.
There are separate terminals for domestic and international operations, though they share the same runways for takeoff and landing.
The airports were formerly called Sahar International Airport and Santacruz Domestic Airport. While hiring a cab to the airport, mention specifically as International airport or as the Domestic airport (more popularly Santacruz airport) . They are a well 15 to 20 minutes drive apart, if you ever need to travel between them. The airlines operate courtesy intra-terminal coaches between the domestic and international airports.
Again there are two terminals each at the Domestic and International airports. While 1A and 1B are domestic terminals, 2A and 2C are International terminals.
The domestic India flights of Air India and the private airline Kingfisher operates from 1A. Terminal 1A is located at your right side as you approach the airport by the main road. Just at the final junction as you reach the airport area, a plethora of airports signboards confront you. Look for the Terminal 1A signboard that points to the right. Tell the taxi driver to take the flyover you can spot a few meters to the right side of this junction. If you miss this you will be by default taken to the terminal 1B. In any case you can walk to terminal 1A (10 minutes walk plus a climb to the first floor lugging along all that baggage....ooff! ).
On the other hand, if you want to go to the city center (or to the suburbs), taxis are available just outside the arrival building. The traffic police queues them up just outside of the lobby of the arrival launch. A bit forward, you can find the three-wheeler taxis (Auto-rickshaws). Note that Auto-rickshaws are not permitted to enter certain parts of the Mumbai (they can take you up to Sion and Bandra in South Mumbai). In other words Auto-rickshaws are not allowed to enter Mumbai City ( the peninsular part of the city ) . So if your final destination is the city center, catch a taxi rather than the cheaper Auto-rickshaws. Their meters don’t display the fare directly. Instead you need to check with the tiny printed card to arrive at the fare in rupees. Roughly the taxis charge 11 times the meter reading and Auto-rickshaws charge about 7 times. (One more complication you’ve to deal with ! But it is a norm in the city and you can manage it easily the next time you catch a city taxi…. ) . Both of them carry this rate card (the fare adjustment chart).
From airport to the ‘city’ the taxi fare is about Rs250 (some 45 minutes drive….). That is a rough estimate for you to get an idea of the taxi fare scenario. Again that goes about 50% up during the night.
The story is not very much different for the International terminal too. However there is a prepaid taxi counter located inside the arrival area. They charge a bit high compared to the metered rates. But this is reasonable a premium, especially if you are arriving at an odd hour in the night.
Domestic India flights from Terminal 1-A:
Air India : Ahmedabad, Aurangabad, Bangalore, Bhopal, Bhubaneswar, Chandigarh, Chennai, Coimbatore, Delhi, Goa, Hyderabad, Indore, Jaipur, Jamnagar, Jodhpur, Kochi, Kolkata, Kozhikode, Madurai,Mangalore, Nagpur, Patna, Pune, Raipur, Ranchi, Srinagar, Thiruvananthapuram,Udaipur, Vadodara, Varanasi, Visakhapatnam.
Kingfisher Airlines :Ahmedabad, Bangalore, Bhuj, Bhopal, Chennai, Delhi, Goa,Guwahati, Pune Hubli, Hyderabad, Indore, Jaipur, Kochi, Kolkata, Mangalore, Nagpur, Srinagar, Varanasi.
Domestic terminal 1B is located at the left as you approach the airport from the main road. A great majority of the private airlines operate their domestic India flights from terminal 1B.
Jet Lite : Allahabad, Bengaluru, Bhubaneshwar, Bhuj, Chennai, Coimbatore, Delhi, Dibrugarh, Goa, Gorakhpur, Guwahati, Hyderabad, Indore, Jammu, Kathmandu, Kolkata, Lucknow, Patna, Ranchi, Srinagar, Varanasi, Vishakhapatnam
Deccan: Ahmedabad, Aurangabad, Bhavnagar, Chandigarh, Chennai, Coimbatore, Delhi, Goa, Hyderabad, Jamnagar, Kochi, Kolkata, Mangalore, Nagpur, Raipur, Rajkot, Thiruvananthapuram, Vadodara, visakhapatnam.
Jet Airways :Ahmedabad, Aurangabad, Bangalore, Bhavnagar, Bhuj, Chennai, Coimbatore, Delhi, Goa, Guwahati, Hyderabad, Imphal, Indore, Jaipur, Jodhpur, Kochi, Kolkata, Kozhikode, Mangalore, Nagpur, Pune, Raipur, Rajkot, Thiruvananthapuram,Udaipur, Vadodara.
IndiGo Airlines : Agartala, Bangalore, Chennai, Guwhati, Goa, Hyderabad, Imphal, Jaipur, Kochi, Kolkata, New Delhi, Nagpur, Pune, Vadodara
SpiceJet :Ahmedabad, Bangalore, Chennai, Coimbatore, Delhi, Goa, Hyderabad, Pune
GoAir :Ahmedabad, Bangalore, Bhopal, Chennai, Coimbatore, Delhi, Goa, Indore, Hyderabad, Jaipur, Jammu, Kochi, Srinagar.