There are many indications to tell whether a bender is a master at their element or not.
For a master waterbender, the ability to swiftly change their element from one state to another and the ability to rapidly switch techniques during combat is an indication that they have mastered it. Keeping the opponent off guard while constantly adapting to the enemy is the way a true master operates. For example, during his battle with Katara at the North Pole, Master Pakku deflected a stream of water away from himself, froze it, and used it to slide behind Katara and counterattacked.
A master earthbender can control larger amounts of earth and physically connect to it as an extension. Techniques such as earth armor, dust cloud, magnetization, and earth bomb can only be performed at this level. Master earthbenders also have the ability to tunnel through the earth, create a wave of earth to ride on as well as use it as a form of transportation, and bend earth that is not physically connected to them. For example, King Bumi used remote earthbending in order to escape his metal cage in Omashu. He was not physically connected to the earth below him, but he managed to bend it using his face.
Firebending masters can control the intensity, pressure, and amount of fire they produce. For example, Azula creates blue flames, which burn hotter than those of a normal firebender. Propulsion, augmentation, wall of flames, and charging are some techniques performed by master firebenders. Other notable abilities include fire comets, missiles, and pinwheels.
As for a master airbender, it is typical for them to be capable of flight and manipulate larger amounts of air. Atypical to their principles, masters can summon nearly lethal moves such as blades and concussors. The ability to create a vortex, a mini-tornado as well as an air spout are also techniques performed by a master airbender.