What is public housing (in America). Why do we need it? Are there not enough landlords?

Is this a form of goverment subsidy? How many public housing units in America are there? If I understand it, I go to work, work hard, pay a full market rent which eats up my monthly income and then some of the money I pay to the goverment goes to pay for someone else's rent? I'm not sure if I'm correct in this umption.