Check the air filter first on an engine that starts and dies; lack of air is the most common cause. Remove the screw that holds the top of the filter compartment, which typically is on top of the carburetor, with a screwdriver and inspect the filter.
Wash a foam rubber filter in warm soapy water if it is dirty, clogged with oil and dust. Replace it if it will not wash clean.
Replace a dirty paper filter.
Troubleshoot the spark plug. First verify that the plug wire is making a good connection with the plug and is not loose.
Use a plug wrench to unscrew the plug. Inspect the electrode on the bottom; if it is burned or fouled with oil or carbon, replace the plug.
Make sure the new plug has the correct gap on the electrode.
Test the gasoline, which can go bad in a mower tank, especially if it has been left sitting for a while. Drain the tank if there is any question; put the old gas in your car, where it will burn without problem.
Refill the tank with fresh gasoline. Use gas without an alcohol additive if the engine is hiccuping or backfiring before it dies.
Clean the carburetor. First remove the air filter compartment and spray an automotive carburetor cleaner into the barrel of the carburetor.
Spray the throttle and choke linkage on the outside of the carburetor to get rid of any grease or debris that might hinder smooth operation.
Inspect under the mower. Clean out any mud or collected grass clippings that might be preventing the blade from turning, putting undue pressure on the engine.
Make sure the blade can turn freely during operation. Remove the plug wire for safety while inspecting the blade.
Look at the muffler to make sure it is not stopped up; an engine must exhaust burned fumes to perform correctly. Clean the muffler or replace it if it is damaged and will not allow air to flow out freely.
Finally, check the control bar on the mower handle, which must be depressed to start the engine, to make sure it is working properly; examine the linkage to the engine to make sure it is connected.