Zach- the guys alluded to, but didn't directly say- Connected in the classic Cub Cadet fashion, an 'automotive' type coil (like what'd be on a V8) won't last long in a Cub Cadet, due to what we call 'dwell'... that's the amount of time that the contact points stay closed, compared to open. Automotive coils are also wound to operate at voltages down to about 8-9 volts, with a ballast resistor wired in series with the coil's primary (low voltage) winding to keep it from burning up. Automobiles use a wire from the starter solenoid to 'bypass' the resistor while cranking. When cranking a big engine, the starter motor can pull available voltage from a weak battery way down, hence, the designers wind coils to fire well down at 8-9 volts.
The coil is a very simple device- it has two windings- a primary, that you run 12v through, and a secondary, that is the developer of 10,000+ volts for firing the plug. To generate spark, you apply 12v across the two primary terminals (the - and + posts). Current flows through the primary, and a magnetic field builds up around the coil, and the iron core. When you disconnect power from the coil, that magnetic field collapses... and when falling, flux(magnetic lines of force) fall across the secondary very rapidly, which induces a voltage in the secondary winding. This voltage is applied to the spark plug (through the coil wire on one side, and the grounded body of the coil on the other). Once the voltage across the gap of the plug exceeds voltage-per-square-inch requirements of coronal physics, fuel/air mix between plug electrodes becomes ionized, carrying 'coil current', which gets very hot, very fast, touching off the mixture.
To control the spark, we simply control current through the coil. The contact points (down in that little box at the other end of the yeller wire)short that wire to ground. That wire connects to the -post on the coil. The other wire connects to + post. when firing-time comes around, the points close (operated by camshaft and pushrod), coil flux builds, then points open, coil fires. Pretty elementary.
The function of the CONDENSER- as was noted by others, the condenser is a type of capacitor... and it connects from the coil - post (with contact points) to the right-side mounting bolt of the coil. It's purpose is to prevent the contact points from being burned-up. See, when the coil flux collapses, magnetic flux lines cross the SECONDARY (creating the extremely-high-voltage pulse), they also cross the PRIMARY... creating a pretty-darned-high voltage pulse, too. Without the condenser to act as a snubber, the undesired pulse causes a spark to jump across the contact points, burnin' 'em up.
And BTW- if that condenser is bad, or the collar or bolt aren't making good contact, you'll not only get continuous bright arcing across the contact points, the engine will start and run at idle... but as engine RPM rises, change in inductive reactance of the coil (with frequency) will cause the engine to misfire like it's got bigtime carbeurator or valve problems.
Just went through that on my 108!
Congrats on rescuing the Cub... rotate the engine slowly, and make sure the piston moves up-and-down... sometimes rods break... then Cubs get parked...