AMR is an advanced voice codec for GSM. A voice codec translates sound (voice) into digital data and vice-versa.
AMR is more efficient than older codecs such as FR, EFR, and HR. Increased efficiency provides increased network capacity as more phones with AMR are deployed. Increased network capacity means more people can use each tower at the same time, reducing failed/dropped calls.
AMR gains efficiency by being "adaptive". Unlike older standards, AMR automatically operates in different modes depending on signal conditions, and continuously adjusts to match the sounds being transmitted.
AMR is used automatically whenever both the phone and the base station (tower) support it. Nearly all new GSM phones support AMR.