When a person says "I am not getting any spark", we have to assume this means the engine cranks whenever you turn the key. Or, it spins whenever you pull the rope.
A mechanically functional (meaning all the internal parts are working, i.e. pistons, valves, etc) engine requires a steady flow of three things, air, fuel, and spark in order for the engine to start and run. These are the trinity, or triad of a happy engine.
If you are in fact not getting spark, after you crank the engine, or pull the rope for a while, you should begin to smell the un-burnt gas passing through the engine. This is a good indication that you have gas flow and air is making it through the carburetor. That is a good thing. It is verification that you have air and fuel, but no spark. If you do not smell this fuel after a bit of cranking you may have fuel problems. Keep in mind that if you smell this fuel it means the cylinder(s) are now flooded and it's time to back off the choke and crank it in the run position.
BEFORE you begin your tests, consider the engines kill system. Modern day safety systems include a kill wire. This will be a single wire going to a connecting terminal on the coil. The wire is small and can be disconnected. Disconnecting this wire isolates the engine from all the possible bad switches, relays, diodes, and wiring setup designed to kill the engine if safety procedures are not followed. SOOO, you may want to run your spark plug test without taking off the engine shroud and disconnecting the kill wire. If you get a quick fix, great! If not you will need to remove the shroud before going through the rest of the tests. You must find and disconnect the kill wire, then begin your coil and spark plug tests again. If the engine starts and runs, then you have a chore in store. You will have to track down the faulty part in your safety system.
Note: it is likely, with a disconnected kill wire the engine may start, but when you turn the key off, or otherwise try to shut the machine off, the engine will keep on running. Just slip the kill wire terminal back on and the engine will die, or wait till it runs out of fuel.
Providing spark to the engine is one of the simple systems and relatively easy to diagnose. Start with the spark plug. You CANNOT test a spark plug (without expensive equipment in your lab). For instance, did you know that it takes more juice to create a spark under compression than it does at normal atmospheric pressure? So just because you have a visible spark from the plug outside the cylinder does not mean you have a usable spark under compression.
The ONLY answer to this end of the problem is to get a new one; even if the current one is not very old. Briggs & Stratton recommend changing the spark plug annually. The spark plug sits in the cylinder and is subjected to compression, high heat, explosive forces, constant vibration, and of course electric shock in the 15,000 volt range. Make sure the gap is set correctly on the new one, typically 0.76 mm (0.030 in).
If you want to know why you should just throw in the towel and start with a fresh spark plug, go here and read this wonderful essay on spark and spark plugs. Village Science Magneto Spark Plugs
So now you have your new plug, properly gapped and it still won't start. Next up is the Magneto or Coil. If you have not already done so, pull the shroud, or fan cover, from the top of your engine and you will see a heavy metal wheel with fan blades. This is the Flywheel/Magneto. It has permanent magnets embedded in the side. As the magnets fly passed the coil, current is generated. Seldom do the magnets go bad, but you should feel a strong attractive force of the magnets when you hold a screwdriver about an inch from the magnets. If they do not seem to be very strong you might have to replace the flywheel. This is a very rare happenstance.
Follow your spark plug wire back to the small metal thingy, mounted next to the flywheel, now you are at the Ignition Coil; or just Coil for short. TWIN cylinder engines have two Coils; one for each plug. A coil generates the 15,000 volts required by the spark plug. The Coil must be a precise distance from the flywheel. This is the Ignition Coil Gap. This gap usually .20 mm to .30 mm (.008-.012 in) from the flywheel. To set this gap properly, the magnets on the flywheel must be right in front of the coil when you set the gap. Check your specs. If the gap ain't right the spark ain't bright.
With the exception of the kill wire designed to stop the Coil from working, the Coil functions entirely independent of the rest of the wiring systems on the machine. If you disconnect the kill wire and the coil is good you will get spark to your engine regardless of the condition of the rest of the entire wiring system. The Coil works all by itself. It does not need the battery or the key switched on.
Inside the Coil are two windings of wire and no moving parts. With shorted (melted together inside) windings you can still get a spark, but it will be weak. With open (burnt/broken) windings, you get no spark at all. There are also electronic components like diodes which may be failing. Coils are subjected to heat and vibration and do fail on occasion. Another coil consideration is the spark plug wire. Back in the day, we would replace all our spark plug wires when we gave our car a tune up; because spark plug wires do fail. The only way to replace the spark plug wire on a mower is to replace the Ignition Coil.
Note: Coils may not fail completely at first.They can be complicit in the situation where a mower starts and runs for a while. As it heats up the coil parts expand and cause the engine to shut down. It does not re-start until the machine has cooled enough to allow the coil to cool. This can be 30 minute to several hours; perhaps it starts again the next day only to repeat the failure.
Short of a well running engine, with no special equipment, the only field-test for a coil and the spark plug wire output is to visually assess the color of the spark. Alternatively, you may use a spark tester. A spark tester gauges the strength of spark by making it jump a bigger gap. This test calls for a gap in the range of 4.2 mm (0.166 in). It's important to note that setting a test gap beyond 5.0 mm (0.200) could damage the ignition components. Briggs and Stratton manuals, for the newer engines, specifically state that you should not test for spark with the spark plug out. I do not know why this is. They specify you should use a spark tester.
B&S notwithstanding (at your own risk), with a new plug attached to the plug wire hold the threaded part against a ground point or wrap some bare copper wire around the threads and fasten the wire to the engine frame or other good ground.Make your shop or garage dark so you can see the spark easily, then crank the engine. The flywheel must spin rapidly (at least 350 RPM). The strength of the spark is revealed in the color. A red or yellow spark is weak and probably will not spark in the cylinder. A blue or white spark is strong and has enough voltage to fight across the spark plug gap even under pressure within the cylinder.
Note: a good strong spark also makes a strong snapping noise as it fires. With the kill wire removed and the Coil gapped correctly, if there is no spark at all with your new plug (try two new plugs in case one is faulty) it may be time to replace the coil.
One final note: On a twin cylinder engine you may have one side firing and not the other; quite common actually. You have the option of switching parts from one side to the other to help diagnose which part is bad. The spark will follow the good part; lack of spark will go with the bad part. It's important to note at this point, IF the safety system is at fault, neither side will spark. That's a clue.
That about covers it for diagnosing and curing a weak spark or no spark problem.