As far as I understand it, First Class will always be sent by Royal Mail and won't require a signature. For the other delivery options it depends on the size of the order. If the total delivery charge (or what the delivery charge would have been, if you have Amazon Prime free delivery) is under a certain threshold then Amazon won't get a discount on it from Royal Mail, and will instead use one of the other carriers; these usually require a signature.
I don't know what the threshold is; also, I may have got it wrong and it's a threshold on the total value of the order rather than the delivery charge.
Just to confuse matters a bit further, I've had large orders (c. £200) which were split into multiple packages. Even though one of the packages was a single book it came via Royal Mail, despite the fact that if it had been ordered individually it would have been sent via another carrier

A further note: if you collect stuff from Royal Mail you have to watch them. I had a card which clearly stated that there were two packets, but the bloke at the office found the first one, handed it over and apparently thought matters were at an end. I had to tell him there was another one, and he had to pull out the card again and look at it suspiciously before grudgingly heading back over to the racks and hunting around. The second box was right next to where the first one had been, label upwards

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