Attempt at a summary of what an engine needs to run:
-You need spark at the proper time, air, fuel, and compression for an engine to start.
-Proper air-fuel mixture is necessary for the engine to run properly. Adjust the main jet and air screw as needed. Main jet is adjusted at WOT and then air screw at idle. Be sure gas is decently fresh and the tank and lines are clean and clear of debris. Be sure carburetor is also clean of any debris. Air filter must be clean and allow proper air flow as well.
-Timing needs to be set right and points need to be clean and properly gapped in order to receive proper spark, as well as power from the battery, coil, and condenser. Coil and condenser must be in working order, as well as a charged battery.
-For good compression, valves must seat properly in the block, valve clearances must be set right, and piston/rings/bore must not be too worn out. Head gasket cannot be blown and head should be an almost perfectly flat surface. Spark plug threads in head cannot be rounded or stripped either.
-The crankcase is also important for an engine to run and stay running. Be sure it is filled with good, clean engine oil, free of any debris. Be sure oil dipstick stays closed and cannot pop open, and that the breather reed valve functions properly. Make sure breather filter element is not clogged up.
A lot of info above and it was kinda confusing to read through it all, hopefully I've summed everything up.