I would like to learn a little more how a radar system works
Firstly there is no such thing as a single radar system, so any answer must be definition general. Il Duce has picked up most of you points, but I would add the following
Primary Radar, without SSR, is simply the display of the reflected radar signal on the Controller's screen. This information may or may not be processed. Without processing, the display is raw, i.e. no speed or heading information, however ATCOs are trained to interpret blips and visually assess relative movement (closure, conflictions etc) between such raw returns and to maintain track identity. As traffic intensity increases in any given airspace, this task becomes much harder; hence the use of SSR or procedural techniques are, generally, used/necessary as a safety back-up.
As to the Controllers actions, this depends on type of service being provided and the classification of the airspace within which the aircraft are flying... for example, within Controlled Airspace in the UK it is quite acceptable for a Controller to ignore a "primary only" return, unless they have specific information that the return is within their airspace, as the regulations to enter Controlled Airspace mandate the carriage of SSR... in otherwords, if the conflictor doesn't show SSR then it cannot be in regulated airspace? But outside Controlled Airspace, a Controller providing a radar service (Radar Advisory) is obliged to pass advise to the pilot to resolve a confliction whether the conflictor is showing SSR or not. (pse accept for brevity, this is a gross precis of the rules but I hope remains accurate. I also note you refer to an incident in SA, my knowledge of procedures outside the UK is limited... but this should give you some basic principles?)