Make $x and you get rental assistance, medical care for your kids, food aid, whatever.
Make $x+1, and you get jack and shit.
There's no reason for it to be that way, in a world with ubiquitous calculators and computers, but that's how it often is.
* If you make $20,000 then your effective income is $32,000.
* If you make $32,000 then your effective income is $32,000.
* If you make $25,000 then you're effective income is $25,000.
This creates a terrible incentive structure where you actually don't want to make more then $20,000 unless you're sure you can make more than $32,000. Well actually more because there's really no incentive to move up to $32,000 exactly.
You have to make sure that when a person makes more money they are always better off for having done so. Having a hard cutoff for benefits doesn't work and neither does a back-off system that matches income (e.g. at $25,000 you get $7,000)
For all the descriptions though, there is a key lack of how to easily describe it. It’s missing it’s catch phrase if you will.
As a first crack at it, let's just give everyone who makes less than say $10k per year a bundle of aid that amounts to the fixed value of $1000 per month. This perhaps an improvement because we aren't ignoring the needs of the poor, but it creates a problem, which is what a welfare trap is. Imagine the graph for this situation. If you're making over $10k per year it's y = x, but if you're making less it's y = x + ($1000/month). So at x = ($1000/month) there's a sharp drop in take-home money. So people who would perhaps like to make a little more money are disincetivized from doing so because they would stop taking home that $1k per month and be exchanging it for a much smaller raise.
The negative income tax proposal attempts to solve this first by just providing cash to poor people instead of various goods that are hard to measure, and then by ensuring that the curve we're talking about never has a negative slope or drop. For instance imagine if you make zero dollars, you're given $1000/month, but then for every additional dollar you make independently, you receive 25 cents less aid. You'll always want to make more independently in this situation even if you don't keep it all.
Most actual examples are less blatant, but let's say our welfare system consists of a 12'000$ welfare for anyone who makes less than 12'000$ in income. If your job pays 11'000$, your take-home pay is 11+12 = 23'000$. If you do very well and get a 2000$ raise, your income is now 13'000$, so you don't get the welfare, and your take-home pay is now 13'000$, meaning you lost 23-13 = 10'000$ of effective income by getting a raise.