There are several ways to approach this common problem.
The number one reason small engines don’t start is that they are out of fuel. Be sure the tank isn’t empty and the fuel shutoff (if equipped) isn’t closed. Once you are certain (or pretty certain) fuel is getting to the carburetor or injection system (turn the engine over and check for gasoline fumes in the exhaust) you should check the ignition system.
Is the engine “turned on”? Look for a switch (and be sure it is in the on position) or a metal lever in contact with the tip of the spark plug (and flip it away from the plug).
If the machine is pull started or an older battery started model and the engine cranks over, check to be sure that all the safety interlock switches are closed. If it is a lawn tractor, the switches might be associated with the PTO lever, seat, clutch/brake pedal and/or gear selector lever. If it is a walk-behind machine, check that the PTO clutch (if equipped) is disengaged and blade-brake handle or other interlock handles are in the correct starting position(s) and their linkages are taut. If the interlock switches aren’t closed, the current required to energize the coil (or other ignition components) won’t be able to flow. In very simple systems, the engine’s on-off switch is nothing more than a ground circuit that acts on the high-voltage side of the ignition.
If the engine still won’t start, there might be a problem with the ignition timing, high-voltage generation or the sparkplug itself. Even if you aren’t mechanically inclined, before calling a mechanic, carefully remove a sparkplug and situate it so the hexagonal portion of the plug and engine block are in contact. Turn the engine over while focusing on the plug’s electrode (formerly located inside the engine). If the plug is firing, you will hear a snap and see a spark at the electrode. If the plug’s electrode is fouled with thick, crusty carbon or other material, replace it with a new one and see what happens.
If the engine still doesn’t start, it might be time to call a friend. The fix might be simple. For example in battery-ignited systems, perhaps the wire that charges the high-voltage source is broken or shorted. In magneto-fired engines there might be a problem with the magneto’s stator — or possibly the magnet slipped out of position. It’s also possible the ignition points (if the engine has them) need adjusting or replacement. The non-start syndrome might also be caused by a leaking cylinder head gasket or any of a number of other less likely possibilities.