Eliminate the obvious if the saw won't start -- the switch is on, the chain brake is not locked, and fuel and bar oil tanks are full. Empty the fuel tank and refill it with fresh gasoline, at least 89 octane, with the proper mix of Stihl two-cycle engine oil if the fuel is old or at all questionable. Pull the fuel filter and pickup out of the tank while it is empty; inspect it and clean it or replace it if it is clogged.
Remove the top cover and take off the air filter. Lift it out from the sides. Wash it in mild detergent and dry it thoroughly or replace it if it is very dirty. Remove the spark plug wire by pulling it straight back and take out the plug with the scrench, a combination screwdriver/wrench that comes with the saw. Inspect the plug. If it is fouled with carbon, clean it with a small wire brush or fine sandpaper or replace it with an identical brand and number of plug. If it is damp with gasoline, the carburetor has flooded. Dry the plug electrode and let both the plug and carburetor air-dry.
Test the tension on the saw chain. It should be tight but loose enough to slide freely. Adjust the tension by removing the cover's retaining nuts with the scrench and turning the adjusting screw beside the bar with the scrench. Tension the chain so it can be lifted from the top of the bar but with the drive teeth still in the bar slot.
Check the bar for any impediments, such as burrs or nicks. Clean the chain slot of oily debris with a fine wire. Inspect the oiler holes in the adjusting area of the back of the bar to make sure they are open so oil can flow freely. Clean the oiling holes in the saw body that feed oil to the bar. Replace the bar if it has places constricting the chain or is otherwise damaged.
Remove the muffler at the front of the saw body and clean the spark arresting screen behind it. Clean all of the cooling fins and other parts, which collect dust and debris. Make sure the small vent in the top of the fuel cap is not plugged.
Adjust the carburetor settings if necessary. If the engine is erratic or has poor acceleration, turn the low speed idle screw (the top one of two on the carburetor) counterclockwise until it runs smoothly. If the engine dies while idling, this screw must be one turn open then turn the idle speed screw (the bottom screw) until the chain begins to turn and back it off 1/4-turn. If the chain turns while idling, turn the low speed idle screw counterclockwise until the chain stops then turn it another 1/4-turn in the same direction.