Ditto that the condenser protects the points. I hope Jerry doesn't mind me modifying what he posted with my own twist.
The points close. The condenser charges up to near battery voltage and at the same time current flows (ramps up) through the coil primary winding and creates a field. Next the points open but the condenser (temporary battery) discharges and continues the primary field current flow for a extremely short time. This allows the points to safely open (minimal arching). The condenser has now discharged. The current flow through the primary winding stops. The primary field flips polarity (the magnetic flux stops but opposing EMF remains). This causes an "inductive kick" to the coil secondary winding which spikes to high voltage on the coil output to fire the spark plugs. The coil primary current now flows in the opposite direction through the capacitor to ground. The field in the primary winding has now completely collapsed and the cycle is ready to be repeated.
If the condenser is missing or open, there is no current flow path that allows the primary field to quickly collapse. That's why the engine doesn't start/run.
