Many 2-cycle engines have problems and are sent to the junk yard before their time. Typical problems include hard starting, rough running, a need to adjust the carburetor during use to keep the engine from stalling, a need to rev the engine to keep it from dying, and not starting at all.
I will assume the user knows to replace the spark plug regularly, to use fresh fuel, and to replace or clean the air filter.
With time and normal vibration, the screws that seal the crankcase from air leaks loosen just a little and air gets into the engine through leaks in the crankcase gaskets.
First check the mounting screws for the carburetor and tighten them. Then go to the screws that hold the cylinder head to the crankcase body and those that cover the end where the crankshaft comes out of the engine. See the yellow circles on this photo of a weed whacker engine. If any of these screws loosen as little as a quarter of a turn air begins to leak into the engine and the fuel/air mixture either is not pushed into the engine on the piston's downstroke or it becomes too lean for the engine to run by pulling in extra air during the piston's upstroke.
After about ten years of use, no amount of tightening on these screws will make a dead engine run. Chances are the gaskets have become hardened beyond their ability to seal the engine. Dismantle the engine completely and install new gaskets. You may not be able to buy the proper gaskets, but you can buy a sheet of gasket material. Use the old gaskets or the engine castings as a pattern to cut new gaskets. Your engine will run like new again.
Step 1: Clean the Exhaust Ports, Too.
Remove the muffler. The exhaust ports are under it. In time carbon residue from the burned oil in the fuel mixture begins to close off the exhaust ports. An engine is an air pump. What cannot get out blocks what needs to get in. Move the piston to the downstroke position (most distant position away from the spark plug) and knock off carbon deposits with a screwdriver. Turn the engine over by hand a few times to blow the carbon granules out of the engine.
Once I used our small 2-cycle garden tiller. After ten minutes the engine siezed and stopped. I had mixed 2-cycle oil with the gasoline, but it was old by the time I was using it. I found advice on the Internet that said to let the engine cool. Then use a wrench with a long handle to slowly turn the engine over. Turn in the same direction, not back and forth. That engine still works very well, although I am sure it suffered a little. Lesson learned: always use freshly mixed fuel.
When a gasoline engine of any kind will sit unused for any length of time, drain the fuel from the tank and run the engine until all remaining fuel in the system has been burned through the engine. This prevents gums and varnishes from closing off small openings.