A leaf blower operates with a small, internal combustion engine, which only requires three basic things to start and run. Air must move in and out of the engine; fuel must move in and out of the carburetor; and the spark must fire at the right time to ignite the fuel. If any one of these three components is missing, the leaf blower won't start.
Check Air Ports
Air moves into the engine via the air filter, which rests above the carburetor. This foam pad removes particulate matter, such as dust, from the air before it mixes with the fuel. If the filter is dirty, air will stop flowing in, and the engine won't start. Wash the filter in soapy water regularly, every 10 to 15 hours, and replace it when it can't be cleaned.
Air moves out of the engine via the muffler, which sits on the opposite side from the carburetor, venting hot gases away from the engine and operator. A metal screen inside the muffler prevents ignited embers from escaping. If the screen, muffler and exhaust port are blocked with carbon buildup, the engine won't start. Clean all of these pars, using a wire brush.
Check Fuel Supply
Fuel moves from the tank, up a hose and into the carburetor; from there, the fuel mixes with air and moves into the cylinder, while excess fuel purges back via a secondary hose. The fuel cycle pulses in time with the crankcase, and if anything stops the fuel supply from cycling, the engine won't start. Clean the fuel tank and fuel pickup nozzles; replace both gas hoses; and remove, disassemble and clean the carburetor if the engine is more than a few years old. Always use freshly mixed leaf blower gas, and dump out any fuel before storing the machine for more than 30 days.
Check Spark Plug
Oxidized or fouled spark plugs will deliver too weak of a spark to ignite the fuel in the cylinder. Spark plugs need replacing at least once a season, or more frequently, depending on how much you use the leaf blower. Check with your operator's specifications for the proper spark plug electrode gap and, if the engine won't start, check this gap with a gapping tool. Clean the boot and wire connected to the spark plug, using a rag, and check for loose connections. A professional should address any other spark problems.
More Serious Problems
In some instances, fixing all of these issues may still not get the engine started on your leaf blower. Compression, or the internal engine pressure, is likely not sufficient to keep all of the moving parts moving in sync with each other. Air leaks around the crankcase and piston often cause most compression-related issues, and often are difficult and expensive to repair. You'll need to have a professional mechanic to inspect and isolate these compression-related problems.