remove the airbox and filter. spray some carb cleaner into the carb throat. close the choke. try to start the engine.
if it fires, you have a fuel problem. start at the gas cap and clean the vent. next, make sure the tank is clean and the gas is fresh and mixed to the correct ratio (if applicable). make sure the filters and valve are clean in the petcock. check the fuel lines for any kinks or blockage. remove and thoroughly clean the carb, useing a piece of wire and carb cleaner. run the wire thru every hole that you can find in the carb body and all its parts. follow up with blasts of carb cleaner.finally, make sure the reed valve is clean, and tight (if your bike has a reed valve).
if it does not fire, then you have either an electrical, or a compression problem. remove the spark plug. insert the spark plug into the spark plug boot. hold the base of the plug against the engine block. turn the engine over as if you were starting it up.
if no spark is present, try a new unused spark plug.
if no spark is present with the new plug, then check the simple things first. alot of bikes must have a good brake and tail light bulb in order to get spark. test the bulbs condition. if bad, replace with a new bulb. if good, then check the magneto. points are a very common item to fail. and most times its because they are dirty, or out of adjustment. clean them with an emory board or a point file. NEVER USE CARB CLEANER! blow them out good with compressed air. an average gap to set the points to is .014"-.018". you can find your actual timing and gap settings for your particular engine in service manuals located at www.mopedriders.org in the moped repair section. coils rarely ever go bad. but, if the magneto is full of rust, every thing needs to be taken apart and cleaned. surface rust on the pick ups for the charging coils, and on the magnets can and will affect spark. make sure the wire from the ignition (ht coil) coil to the spark plug is in good shape. check the spark plug boot for continuity, especially if it is aresistor type boot. and if it is a resistor type boot, do not use a resistor type spark plug. lastly, check all wire connections to ensure they are clean and tight. on puchs, make sure the horn is hooked up.
if it has spark, then you have acompression issue. all 2 strokes must have at least 120 lbs of compression to run. so use a guage to test the compression. if less than 120 lbs, you must find where it is going. the usual places are, rings, head gasket, cylinder base gasket, intake manifold (where it mounts to the cylinder), or crankseals. a leak down test can be performed by any reputable lawn mower, auto, or motorcycle repair shop to tell you exactly where your compression leak is and how bad it is. replace or repair the bad parts.
if compression is higher than 120 lbs, pull the exhaust system off and make sure the port and the pipe are clean. these two things will prevent starting.