engines are not the stuff of nuclear power plants or the female psyche.
the "gas" engine needs a very few things to run.
2) fuel (the right amount, at the right time)
3) spark (at the right time)
alrighty then, you've got a 2 stroke, easy peezy.
(check to make sure you have fuel, and its connected and "on". dumber things have keep me at the dock, it ain't personal.)
you need to check two things, no matter what the order.
A) Are you getting fuel thru the carb?
B) are you getting spark?
these can most times be accomplished by taking out a spark plug.
Is the plug a bit "moist", not wet? Or, is it dry.
If its dry, fuel isn't going "thru" the carb., find out why. Check fuel supply, lines, filter.
Tap the side of the carb with a plastic handle of a screwdriver. Don't beat it to death, just tap it. Give it another try. If you have fuel to the inlet of the carb, it may need to be rebuilt.
If its wet, then assume fuel is getting thru, maybe far too much. dry the plug(s) off, or replace them. Check the "choke" on the carb, make sure its not sticking.
If its moist, (lightyly damp) then fuel is getting thru, on to plan B.
Spark. With the plug(s) out of the engine, connect one to the wire and ground it. The "grounding" can be having a clip from the metal base of the plug to the block, or if you've got a kid, have him hang on to it. and pull the cord. If the kid jumps, (Its not gonna kill him, it builds character... I wouldn't try this with the wife, she'll beat you till she gets tired) then you got spark. If you see a nice blue arc between the electrodes, you got spark.
If not, find out why. (make sure all electrical connections are tight, replace the plugs with new, or known good ones)
Look at the plugs. Is the business end, grey, brown, white, or black, is there any crud on the end? that may be a clue as well.
Thats a good start, there are a bazillion possibilities below these options, let us know whats what, and we'll take a stab at it.
As a side note, my neighbor is the regional manager of the pharmacies at a very large chain.
One day his lawnmower wouldn't start, he pulled on it for a good half an hour. Then he comes out of his garage, and beats the paint off of the thing wiht a shovel & drags it out to the curb. He said he's leaving it there as a warning to all of the other lawnmowers on the street.
I bent over, turned on the gas, pulled it 3 times. It runs fine. Poor besterd can't even look at me.