Simply put the problem will be in one of three categories: spark, fuel, or compression (including valve timing). If it was running OK when it was last shut down, it is unlikely to be compression, and you can normally feel it when you pull the rope anyway. You can check spark by removing spark plug, attaching plug lead, grounding plug body, advancing the speed control to maximum, and looking at the plug gap while pulling the rope. If it doesn't spark, you've found the problem. If it does spark, it's likely to be fuel not reaching the combustion chamber. You check that by putting a modest amount of fuel in through the spark plug hole, then trying to start the mower. If it starts and runs briefly then stops, your problem is fuel supply: tank, hose or carburetor.
