Engines basically don't run for one of two reasons:
If you pull off the spark plug wire, and hold it near the engine (mayb
1/4"), do you get a spark while cranking the engine?
Remove the spark plug (special wrench/socket needed), is the gap either
excessive ( > 0.030"), or bridged with carbon fouling? If either,
clean, reset gap, and put the plug back in.
If you are getting fuel, you should be able to smell it. The plug
should be wet, if there is no spark.
How about the air filter? If it is the sponge type, it needs to be
washed out now and then. If it is the paper kind, they have to be
replaced every season or so. Take a good look at it. In a pinch, use a
whisk broom to clean off the grass clippings, crud, on the surface of
the air filter. Put it back in.
That should give you a good start. Also, the manual that comes with
your mower should give basic maintainence tasks that you can do.
When you turn the engine over with the pull cord, you should feel some
compression, unless the engine is very worn.