Yes the compression is vital. What the engine does is get the fuel and air from the carb. Then the piston goes up and compresses that air/fuel mixture enough so it ignites shoving the piston back down to have it repeat over again. When the compression drops the piston can't compress the air enough and it will have problems igniting the fuel/air mixture, so you will see a big drop in power, idle problems, tuning problems, and finally an engine that won't run. You can check compression by turning over the flywheel, you should feel a definite cycle from easy to turn to harder and harder then it should give out.

Good news is most of your engine is fine, all you need to replace is your piston and sleeve which will cost you around 40 dollars on a non race engine, up to 90 or more on a race engine.