What New York's Affordable Housing Programs Actually Require
Affordable is defined against a regional income measure, which is why the label so often confuses.

Housing described as affordable in New York is priced against a formal income standard rather than against any general sense of what is cheap. Understanding the mechanism explains most public confusion about it.
The income measure
Programs set eligibility and rents by reference to area median income, a figure published for the metropolitan region by the federal government and adjusted for household size.
Because the measure covers a region wider than the city, and because incomes vary enormously between neighbourhoods, a unit affordable at a given percentage of that measure can be well beyond the means of the surrounding area while still meeting the program's definition. This is the source of most local objection, and it is a real effect rather than a misunderstanding.
Bands
Programs target bands — deeply affordable units at low percentages of the measure, moderate and middle income units higher up. A development's affordability is described by which bands it serves and in what proportion, not by a single label.
How units are created
Some come through mandatory requirements attached to zoning changes that increase permitted density. Some come through voluntary participation in programs offering tax benefits or financing in exchange for affordability commitments. Some are built directly by public and non-profit developers.
Each route carries different requirements about how long affordability lasts, which is among the most important terms and among the least reported.
The lottery
Units are allocated through a housing lottery run by the city, with applications made through a central system, income verification, and preferences that may apply to certain applicants.
Odds are long because demand vastly exceeds supply. Applicants should apply to everything they qualify for, keep documentation current, and be aware that the verification stage is where most applications fail.
Reading an announcement
When a project is announced with a number of affordable units, the informative questions are: at what income bands, for how long, and how does the count compare to the additional density granted in exchange.