The breaker point(s) are part of the ignition circuit. It's basically a switch that is normally closed, and then opened at a precise time by a cam lobe on the crankshaft. This sudden opening of the points is what creates the spark, by initiating a pulse of current through the ignition coil. The energy comes from the magneto, essentially a rotating magnet that passes by a pickup coil and induces a pulse of current. There is a capacitor (sometimes called a condenser) connected across the points to prevent arcing as the points open and close. It's not uncommon for points to wear so that they don't make good electrical contact, and for the points to get pitted due to arcing. They are not expensive or difficult to replace, and the set usually comes with a replacement condenser as well. Let me know if that info helps at all.
The magneto will have a "kill" wire that inhibits the ignition when it is grounded. This is usually connected to the key switch or a shorting bar on the throttle as a safety control. It has to be un-grounded for the magneto to function. Just as a caution, after a near miss with a live magneto on a Cessna 150, I don't trust the magneto "switches" to prevent the motor from starting when you've got your hands in there (rolling on belts etc). I always disconnect the spark plug wire as an extra precaution when turning the motor over by hand.