The term ignition refers to whatever mechanism is employed to cause combustion in an engine. Spark, glow plug, and compression are three common methods of ignition found in internal combustion engines, although mopeds almost universally use spark ignition. There are two main types of spark ignition found on mopeds.
In this system, a set of breaker points works together with an ignition coil, a capacitor, and a high-tension coil to deliver spark at a set timing. An ignition system utilizing points will deliver spark at the same interval regardless of RPM. Some of the pitfalls of points is that the contacts points themselves can get dirty, and if they are dirty, they will not function, and your moped won't spark. Another problem is that over time, they gradually wear away altering your timing, and will eventually need to be replaced. On the other hand, a new set of points are relatively cheap, and fairly easy to find for most moped brands. Lastly, a big problem with points is that they function mechanically. There is a point where points simply cannot keep up with the speed of the engine at high RPMs. However, this is not a problem for most moped riders/tuners as those RPMs are pretty exceptional (at least on Puch).
CDI stands for Capacitor Discharge Ignition, and it differs from points in that it is an electronically controlled system. Therefore, it avoids the problem of wear and tear inherent in a points-based ignition. A CDI ignition also has the advantage of using a CDI 'box' which controls timing. Various boxes are available to give your engine different ignition timings throughout the RPM range. This can result in ignition timing that is close to ideal throughout the RPM range if you get the right box. CDI ignitions also tend to give hotter, more consistant spark, and require less maintenance than a points-based igntion.