When signing up for Amazon EC2 I had to activate the service using my phone. I clicked a button that said "Call me now", then amazon displayed a PIN on my screen which I then had to key in my phone once I received a phone call from them. Once the call was completed my screen automatically updated and said I was approved.
Which service allows Amazon to do this? I know they're Twilio and other services but I'm wondering if anyone knows which service/API they use.
You could achieve that with Twillio. I just recently played with their API (through the PHP library they provide) and they make these kinds of things trivial.
So in this case, it sounds like you want Gather: http://www.twilio.com/docs/api/twiml/gather
On top of the their markup, you can constantly poll for the status of a call, etc. so you can update your own application in real time, e.g. to reflect that the call finished or if the input was correct or incorrect.