I was trying to find out info about this and somehow couldn't find any one site that spelled it out clearly, so I am doing this here. Roaming charges are freaking insane, so if you want to really save on that this is what you do:
1. Signed letter from your hotel confirming that you are staying there.
2. A passport-sized photograph
3. Details of a local contact in India (Address, Full Name and Contact No.)
4. Photocopies of your Indian Visa and your Passport (the page with all your details)
I only had the first one when I went to the Airtel centre as I didn't know any better. Ended up making 2 additional trips, one to get a copy of my passport and the other to find a photo studio.
That's it. I used Airtel since that was what the hotel recommended, got a cab to drive me to the nearest centre and got it done, you will get your SIM card immediately. It includes data plan as well. I paid around 750 Rupees in total for 650 MB of data and a minimal voice plan thrown in.
The SIM card I got is valid for a year without usage.