There are basically three elements involved in getting a piston engine to run: spark, fuel, and compression. You need to check all three.
First, to check spark, you remove the spark plug, attach the plug lead to it, and lay or clamp it onto an unpainted metal part of the engine. Usually a cylinder head cooling fin is good. Set the speed control in the starting position, and pull the starter cord while watching the spark plug electrodes. Expect to see a blue spark between them, once each second revolution of the engine. If there is no spark, there will be no action from the engine. Tell us about it and we'll discuss how to find and fix that fault.
Second, if you have spark, put just one spoonful of petrol into the engine through the spark plug hole, then install and connect the spark plug, set the speed control for start, do not operate the choke (leave it pushed in, which is open) and pull the starter a couple of times. Expect the engine to start and run for a couple of seconds then stop. If it does, your spark and compression are not the issue, the problem is no fuel is getting into your intake pipe from the carburetor. Report this to us, and we'll talk about carburetor diagnostics.
Third, if the engine has spark, but won't run when you provide it with fuel, you'll have to check the compression. This is usually done with that Briggs engine by disconnecting the spark plug wire, and turning the clutch housing backwards by hand. You should find at one point it becomes harder to turn. Rotate it forward a half turn, then spin it backwards with your fingers, so it bumps against that point of resistance. It should bounce off that compression pressure, and rotate forward. If it does that, chances are it has enough compression to start and run.
Please report the results of these tests, and we'll try to figure out what the results mean, and what to do next.
Incidentally, both of your outer chain's sprockets are worn, the smaller one fairly badly worn, so the chain is probably worn out (that is what causes the sprocket wear).
