Pull the starter rope through. Does the engine spin freely (bad news, repair shop time) or does it take a healthy pull and you can feel the resistance. Hopefully you can hear the "sucka sucka" sound of the air being sucked in the carburetor.
Not sure? Remove the spark plug. Take a look at it, if it's black and sooty or wet, this is your most likely problem. Is it absolutely dry? Replace it with a new one if you're not sure, but be sure to gap it properly first. Maybe you're not getting any fuel. While the plugs out, lets do a quick and dirty compression test. Stick your finger over the plug hole firmly (or your thumb if you have small hands) and with your body parts well clear of the prop, hit the electric starter, or have a friend pull the motor through. The compression should blow your finger off the plug hole. (If not it's probably repair shop time). If the compression is good, and the plug looked okay before you need to check for spark. Clip the spark plug cap over the plug, and rest the plug on a metal part of the engine. Either pull the motor through quickly, or hit the starter and you should see a definite spark popping consistently from the plug. It should be bright bluish white. IF it’s an orange color your spark might be weak. If there's no spark, make sure the kill switch or safety switch isn't off.
If you have compression and spark, the last thing to check is fuel. Take a small amount of fuel and squirt it into the spark plug hole - quickly put the plug in before the fuel evaporates. Put on the plug wire and crank it over a few times. If it pops to life for a couple seconds and then quits, you know it is a fuel supply problem. If you can crank a half dozen times or so with no effect you may have squirt in too much fuel. Engage the throttle wide open and pull the motor through a half dozen times. If it still doesn't start you most likely have either a weak spark that doesn't work under compression (check the coil or most likely the condenser, clean the points).
If it fires briefly and you can't get it to start again, pull the plug, squirt some fuel in again, insert plug, crank. If it pops again but doesn't run, you have a couple options. The best is to pull the carb off and clean it. The lazy solution is to take off the air cleaner and squirt some mixed fuel into the carb Venturi. Crank the engine till it fires, and keep squirting fuel into the venturi until the engine will keep running on it’s own. Often times when engines sit, they get a bit of gunk in the carb and this is a quick way to solve the problem without pulling the carb.
This will solve about 90% of your problems. A couple of odd things to look for are a sheared woodruff key. This lets the flywheel spin on the shaft, the magnets aren't oriented correctly, and the motor goes out of time. Sometimes the key bends without breaking, throwing the timing off just enough to keep it from starting. Tight belts let the prop keep things turning easier, so make sure you check those as well.