The chainsaw engine needs gas, compression and spark in order to start. First, make sure there's fresh gas in the tank--old gas accumulates water and burns poorly. Replace the fuel if it's old. Add fuel stabilizer to the gas when you refill the tank to help keep the fuel fresh.
If the fuel is fresh, check for and replace cracked fuel lines. Air enters the carburetor instead of fuel when the fuel lines have cracks. If the fuel lines are okay, you may need to rebuild or replace the carburetor because a dirty carburetor won't supply the engine with fuel.
Next, disconnect the spark plug wire and remove the spark plug. If you see deposits or corrosion on the electrodes, replace the spark plug.
Finally, to check for compression, remove the spark plug, press your thumb over the opening and pull the starter cord. If you don't feel air pushing from the cylinder, it's likely that the piston rings need to be replaced. (For an accurate compression reading, use a compression gauge.)