You need to figure out what part of the equation is not working. As stated above, there are three basic things (even though some would consider it four) you need for an engine to run: fuel; air; fire. A fourth could be considered compression, but to me that falls under the air category, so we'll leave that be.
The first thing to do is pull the air filter off of it and ensure it's not clogged up. Just leave it off during the rest of your testing, as it will make it easier. If it is clogged up (black, grimy, dirty), try to get it started without it on there and see if there is any difference.
If that fails, pull the spark plug wire, then spark plug out. Test the spark plug wire by holding it close to a metal part as you crank the engine. All settings must be in the "run" mode, because if the auto stop is engaged or if it is in the "stop" position, you won't get any spark. You may want to hold the lead (or have a 2nd person help you) by holding it with an insulated pair of plier or if you can get it wedged in there some how to be able to see it and have it close enough to jump the gap. The electricity coming out of it is not enough to hurt most people (ie: If you have a pace maker, don't let it get you!), but is enough to know you've got juice there. If you see it jump the gap, do the same thing with the spark plug in place. If no spark at the first phase (wire), you'll need to get a new magneto for your mower. They very rarely go bad, so I wouldn't expect this. Obviously, if there's spark at the wire, but not across the plug, you'll want to replace the plug.
If the spark plug is working, next thing to do is squirt just a little bit of gas into the cylinder, then put the spark plug back into place, along with its wire. If it starts up and runs, even for a brief few seconds, your issue is fueling. The carb probably is plugged up so is not letting it run correctly. This, by far, is probably the most likely scenario. The easiest and cheapest way to fix this is to buy a new carb for your mower and install it. If you are wondering how I can consider this "cheaper", consider what your time is worth and ensure you've included that in your cost equation. I personally work for $80/hr ... a $30 carb is well worth my time versus trying to fix it. Chineesium carbs are fairly cheap off of Amazon or eBay, as long as you can find the correct one.