Despite our best efforts to the contrary, there are times when a piece of equipment refuses to start. I'd like to expand a bit on what Defender968 posted above. (He was spot on BTW)
Most gasoline-powered equipment has three basic requirements to run: Air, fuel, and spark.
The air is the easiest to check & rule out. Inspect the air filter and clean as necessary. Peer into the carb while the filter is off to insure there are no obstructions. Operate the choke controls and observe (there would be a metal plate that rotates when this control is engaged, blocking off the airway.) Also operate the throttle trigger and insure it's not sticking or that the linkage is disconnected. One could leave the filter and shroud off for the next steps, but do reinstall these before normal operation.
Fuel (more specifically, fuel and air under compression) is the next thing to check. A quick & dirty compression check is to hold the saw up, grasp the starter handle, and let the saw drop. With "good" compression, the saw should hang, or very slowly drop. (blup.......blup......blup etc) If the saw freely drops or offers little resistance, something is worn internally. (Blupblupblupblup) Of course, if the internals are locked up, the rope wont pull, and the show is over until those issues are addressed.
Some pro-grade saws have a compression release device on the top or side of the cylinder. It's function is to allow for an easier pull when starting. If this device is engaged or stuck open, there is no hope of getting good compression, and the engine will likely not run. Rule this out.
A compression gauge may give a closer idea what's going on inside, but these diagnostic tools vary in application. A saw has a very small volume, so an automotive gauge may give falsely-low readings on a small engine. In a nutshell, 180psi would be outstanding, down to about 130-140 psi is serviceable, and below 125psi indicates the chestnuts are roasted. If the compression is very low, one could remove the muffler and carburetor, then slowly rotate the crank and observe the condition of the piston, rings, and cylinder bore. If things appear pretty smooth, or just lightly scuffed, that is good. If it appears to have spent it's summer vacation in Mogadishu... bad news & steel yourself up for a visit to the New Engine Store.
If the saw has an unobstructed airway, holds good compression, and has known good fuel, the only thing left is to check for spark. Remove the spark plug and inspect it (or if in doubt, replace with a new one) it should have only light brown deposits on it. Dark oily deposits indicate an overly rich condition, and one that is very white indicates a lean condition. "Reading" a two-stroke plug is somewhat voodoo science, so I default to swapping in a fresh part.
While the plug is removed, it's a good idea to pull the engine through a few times to purge any accumulated fuel. One might have to invert the saw, and pull, just to insure that no fuel has pooled up in the valley below the crankshaft. If one has flooded the engine badly, it may take several hours to clear it out. (Pull awhile, let it sit, repeat as necessary.) Once things are sufficiently dried out, confirm that the kill switch is in the "run" position, attach a known good spark plug to the lead wire, ground the plug electrode against the cylinder (this may take several hands, or rigging up some apparatus) and pull the starter. One should see a bright blue spark at each revolution. As Def968 advises, do not become part of the ground path. Letting your spouse or children hold the plug is an option
However these folks may be caring for you in your old age, so memories of Grandpa's workshop should probably not include high-voltage. Failure to achieve a nice fat spark indicates something is amiss with the electronics (The kill-switch is in the 'run' position, yes?) Older saws & tools had points & condenser that could fry, wear, or come out of adjustment, but new equipment is pretty much a case of it's working or it's not.
Some equipment has an air-purge bulb ('primer') that forces the air out of the carburetor passages. Like anything rubber or plastic, it will degrade over time. If this is inop or leaking, it's fairly easy and inexpensive to replace. Same applies for fuel lines. These are also sacrificial parts, and especially so with ethanol blended with our gasoline.
If the saw runs, but seems to want to hold it's revs, or reluctantly idles back, this is not a new-found cruise control... It indicates an air leak somewhere which needs addressed ASAP. Your expensive tool is heading towards becoming a Chernobyl replica at a double-time. See "New Engine Store" above.
That's about it for two-stroke engines, at the very basic level. Snow-mos, watercraft, dirt-bike/quad engines etc... may have advanced sub-systems such as fuel injection, variable port timing ('power valves') and other features to increase performance, reliability, and control of noise or emissions. These all add complexity, and have their own specific procedures for troubleshooting.