Depending on the style of ship, company, and other factors, they are called "the third" or "the fourth," usually stand a watch and sometimes assist the third mate in maintaining proper operation of the lifeboats.
Generally speaking, third assistant engineers are tasked with controlling the main propulsion systems, water systems, purifiers, air compressors, sewage treatment plant, fresh water generator, actively involving in bunkering operation, record keeping of fuel tanks sounding as instructed by chief engineer and anything else tasked to them from up in the chain of command.
The third assistant engineer is in charge of keeping the plant in operation, with command being passed down and being delegated by whomever is on watch.
They are trained to be competent in a wide variety of fields such as HVAC (heating, ventilation, and air conditioning), electronics, hydraulics, engine room auxiliary machinery and the operation and maintenance of both steam and diesel propulsion plants.
As with all seafarers, the third assistant engineer is an integral part of the emergency response with critical technical knowledge.