There are three main ways low-cost two strokes can control the induction process: a piston-controlled intake port at the bottom of the cylinder bore, to let gas into the crankcase, a reed valve controlling an intake port permanently connected to the crankcase, or a rotary valve controlling an intake port into the crankcase. Rotary valves are fairly expensive and are mainly found on relatively high speed two strokes, such as motorcycle engines. Reed valves cost slightly more than piston-controlled ports. The lowest-cost two strokes usually use piston-controlled ports.
A piston-controlled intake port is opened and closed by the skirt of the piston, not the top edge above the rings, and it only opens briefly at the top of the piston's stroke. At that point there is a vacuum in the crankcase because the piston has moved up to the top of its stroke with the crankcase sealed, thus increasing the internal volume of the crankcase. As soon as the piston begins to descend down the bore the intake port is shut off by the piston skirt, and the gas in the crankcase starts to be compressed due to the crankcase volume decreasing. At the bottom of the stroke the top edge of the piston opens the transfer port which connects to the crankcase. The transfer port allows the compressed gas in the crankcase to be transferred into the cylinder, ready to be compressed as the piston moves upward.
If your engine won't start there are some basic checks to make: is there spark, is there good fuel in the tank, does the fuel flow readily into the carburetor, does the compression feel normal. Before assaulting the carburetor and possibly making things worse, it pays to identify the general cause of the problem. If you have spark, fuel is entering the carburetor, and you are confident that compression is normal, you could try squirting some fuel into the engine (through the spark plug hole or the carburetor air intake) and see if it starts then. If it starts but then stops, it's time to take a look at the carburetor. If it still doesn't start, you most likely have loss of compression, or less likely, a problem with the speed control or governor. If it has spark and priming the engine with fuel as described does not make it start, checking the decompressor might be a good idea - they seem to give a fair amount of trouble.