How is life of aircraft structure counted?

It's counted in flying hours and flight cycles. The primary concern is flight cycles.

A flight cycle is a take-off and landing. Airlines keep track of the Flight Time (hours) and Flight Cycles on the airframe and the same for the engines. Flight Cycles are primarily important for tracking aircraft metal fatigue and the time on the Landing Gear. The parts on the gear have a Safe Life Limit calculated by the manufacturer based upon fatigue tests and stress analysis. This limit represents the amount of cycles it can be flown before it will begin to develop fatigue cracks. Once the landing gear part reaches that cycle limit, it has to be removed and scrapped. The manufacturer analyzes the gear using various loads such as landing loads, taxiing loads, braking loads, turning loads and calculates a fatigue life for the part. They usually then divide this calculated fatigue life by 4 and this becomes the Safe Life Limit. Dividing by 4 is a safety factor to allow for variations of manufacture and environmental unknowns and statistical scatter in their data. The Airframe is also tested in a fatigue test for 2 or 3 times the time it will be in service. Many airframes have a 90,000 flight cycle limit. Some more. Helicopters will be less. This airframe life limit takes into account air pressure on the fuselage as it cycles from ground to high altitude. Also, a short test flight is counted as 1 flight cycle.

As an example of the difference between hours and cycles, the Aloha Airlines Boeing 737 that lost part of its fuselage had only a little over 35,000 hours on the airframe, but the aircraft had almost 90,000 cycles on it.