Let's say a unit is "50% AMI" somewhere with an area median income (AMI) of $100k. You might think, and I've seen a bunch of people with this confusion, that units would rent for 50% of $100k: $50k/y ($4,170/month), but it's much cheaper than that.

Affordable has a very specific meaning in this context: spending no more than 1/3 of your income on housing. So you might think 50% AMI is 'affordable' to someone earning at the 50th percentile: 1/3 of $100k, or $33k/y ($2,750/month), but it's cheaper than that, too!

Instead, 50% AMI means someone earning 50% of the AMI would be spending 1/3 of their income on housing. This is 50% of 1/3 of the AMI, and in this case that would be $17k/y ($1,390/month).

Now, I don't think affordable housing solves everything: if it were widespread I think it would be exploited and it can be a distraction from just getting a lot of units built, but the term "affordable housing" is actually a decent operationalization of whether housing is something regular people can afford.