What do you call a system that sets aside a share of total places, seats and so on for a specific group?
The most common name for a system like this is “affirmative action.” We can also speak of this as a “quota system,” but the term “affirmative action” should be used so long as the group is being given its set share of places because it has been discriminated against in the past.
In countries like the United States in which there is a racial minority that has been abused in the past, systems like these are somewhat common. The systems work to try to make up for past discrimination by making it easier for members of the minority groups to get jobs, places in colleges, and so on. This is called affirmative action. When the affirmative action program actually sets explicit numbers for each group, it is known as a quota system. Quotas are illegal in the United States. Instead, schools tend to take steps short of setting quotas. The legality of these steps is hotly debated in the US where a case about affirmative action is being decided by the Supreme Court this term.
The correct term for this sort of a system, then, is “affirmative action” and, perhaps more precisely, “affirmative action using a quota.”