Go to Autozone or possibly Walmart and pick up 2 things: Berrymans B12 SPRAY and a can of starting fluid (various brands). Spray the heck out of the carb area with the B12. You don't have to pull the carb but it is a good idea to take out the air filter and shoot some down that way. Then, try to start it. If that fails, shoot the starting fluid in either the same area (air cleaner) or for some mowers, down the muffler will work too. That SHOULD start the mower.
If it starts but won't run, your gas is bad. You don't have to throw it out completely, but get it back into a 5 gallon can and dilute it with fresh gas. At any rate, get fresh gas into the mower and it should run. If all that fails you have to look into the plug or air issues, but I've never had those issues on a mower. And I've pulled and cleaned or rebuilt carbs on probably 8 mowers and a half dozen other small engine items.
Every year at the end of mower season, change the oil, run the engine until it runs out of gas, spray the carb with B12 and every other year put a new plug and air filter in. In the spring, spray with B12 again and possibly starting fluid, and use a gas additive to keep your gas fresh all year.