Exact implementations vary, but the general idea is that developers of multi-unit housing projects are encouraged to set aside a certain percentage of their units, generally raging from 10-30%, but sometimes even more, as "affordable housing" units.
In other words, some proportion of the units are under rent controls to the point where they must be rented (or sold) at a loss by the developer. Sometimes the schemes are voluntary and give developers density bonuses, sometimes developers can pay a fee instead of setting aside units. The exact proportion of units that must be set aside and the loss that developers take on each unit also varies. As you can imagine, I'm not in favor of this system, but it's a complicated issue.
Inclusionary zoning (IZ) is a relatively new concept, first implemented in the 1970s, to combat the growing problem of residential segregation of classes and races, whose origins are interesting and, I think, germane to the conversation.