Understand what a capacitor does. We all know that a capacitor "stores charge" but what does that mean in our practical application? In a circuit, a capacitor resists(not like a resistor though!) a change in voltage across its two nodes. Connect a capacitor in parallel with a battery. The moment before connecting the battery, the voltage across the cap is zero. The very moment after connecting the battery, the voltage across the capacitor is still zero! Of course, the voltage across the cap soon rises to be equal to the voltage of the battery.
We can also say that the voltage across a capacitor will always change continuously(i.e. it will never 'jump')
Similarly(but harder to explain without dealing with calculus) the current flowing through a coil(or what we in the biz call an "inductor") will always change continuously
The condenser prevents arcing as follows. The condenser is wired in parallel with the points. When the points open, the voltage across the condenser can not change instantaneously. Because of this, the voltage will slowly rise across the capacitor as it charges. Because the points are wired in parallel with the capacitor, the voltage across the point gap will also rise slowly, preventing an arc.
There is an easier way to think of the condenser/capacitor's operation: As the points just start to open the flow of current will start to jump the gap, but as this is happening the resistance in the points circuit is also going up. The capacitor is at zero resistance at this instant so the current will flow into the condenser rather than jump the gap.(current always takes the path of least resistance) As this is happening the points will get so far open there is not enough voltage in the system to jump the gap, so no arching takes place. When the points close the capacitor drains it current back the ground thus setting its resistance back to zero, making it ready for the next cycle.
Consider a magneto without a condenser:
Before the points open, the flywheel(which is also a spinning magnet) is inducing a current in the internal ignition coil(the top coil, in the above picture). Since the points are closed, this current is going directly to the ground.
The points just begin to open(now there's a very small point gap). The current flowing through the internal ignition coil remains constant(Faraday's Law), but so does the current flowing through the primary winding of the external ignition coil - but there isn't any current flowing through the primary winding of the external ignition coil! Therefore since it has no place to go, the path of least resistance for this current in the internal ignition coil is to jump across the very small point gap to ground.
This arcing has two results: 1) there is less energy(if any at all) left to create a spark in the spark plug and 2) the arcing causes corrosion(which has a very high resistance) on the surface of the points that are supposed to touch.
Why is number 2 bad? because if current can't flow through the closed points into ground, then current can't get established in the internal ignition coil!
Consider now the magneto with a condenser:
Initially, nothing is different.
This time when the points just begin to open, the current flowing through the internal ignition coil does have a better place to go. Instead of arcing across the just-open points, most of the current flows into the condenser. Some of it starts flowing into the primary winding of the external ignition coil as well(but that's a tail of two coils for another post!). By the time the condenser is "full", the points have opened so far that the voltage across the condenser could not possibly jump this gap. The charge stored in the condenser ends up resonating with the internal ignition coil(in a CL-resonant circuit, again that's for a different post).
If the condenser is shorted internally it won't matter that the points are opening and closing because the current will always be flowing to ground through the condenser. If the current doesn't go on and off through the coil you get no spark.
To test the condenser you have to completely isolate it from the circuit and test it with an ohm meter. If it is shorted it will show zero ohms. If it is good it will show a range of ohms as it slowly fills then ends at open circuit, infinite ohms. Use a jumper from the center wire to its case and it will discharge and you can repeat the test. Heat and the high voltage produced at high speed will eventually kill the condenser.