If a snow blower won't start, or sputters and dies, check the gasoline tank. The majority of snow blowers are powered by gasoline, and they can't run on empty. If the gas tank is empty, refill it. Sometimes issues with the gas tank can be related to the type and quality of gasoline being used in the snow blower. Generally, gasoline that's used in snow blowers is sold specifically for the devices, and is formulated for the winter season. This gasoline produces more vapor, and makes it easier for the engine to start. Also, if the temperature dips below freezing, and there is moisture in the gas line, the line can freeze if the tank doesn't contain correctly balanced fuel. All snow blower fuels should include an antifreeze additive that prevents this from happening. Replace the gas in the fuel tank, if necessary.
Check the Spark Plug
If the fuel in the snow blower appears to be effective, the problem is likely elsewhere in the device. Many problems can be detected by taking a look at the spark plug. After pulling the cord on the starter a few times, pop out the spark plug, and take a look at the bottom. If the bottom of the spark plug is wet with fuel, that indicates that the snow blower either contains bad fuel, or isn't sparking properly. Devices are made to test the spark, but they can be dangerous to operate. The best bet is to find a professional to test the spark plug, if this is suspected to be the problem. If the spark plug has no fuel on it, the carburetor may be the problem. Usually, this issue can be remedied by removing the carburetor from the snow blower, and cleaning the fuel deposits off of it.
If everything else checks out, the problem with the snow blower is most likely a more serious engine problem. The only way to really properly (and safely) diagnose an engine problem in a snow blower is to either take it completely apart, or hook it up to a diagnostic machine. Both options are best performed by a professional. The gas tank poses serious risk of fires and explosions, particularly if the cause of the problem is unclear.