In Missouri, at least, it can happen a few ways. During a divorce or custody hearing, a judge will order if and how much child support a parent should be paying. If there is no divorce or custody hearing, the custodial parent can go to FSD and request they bring a child support order against the non-custodial parent.
If those payments aren't made, the custodial parent can push for garnishment of wages - this would include income from work, tax refunds etc. Anything not paid will just build up into a balance owed. Once a person owes a certain amount, the state can then go further with trying to force payment such as revoking a driver's license and having the person arrested and jailed for non-payment.
Sadly, at least in my case, most of this doesn't make a difference. My children's father currently owes me nearly $10k. The child support caseworker contacted me months ago and asked if I would cooperate in legal proceedings against him which I agreed to. I have heard nothing since and when I call to ask, they tell me they have my case marked to not be reviewed again until November 2018. He works under the table as a maintenance man for the trailer park he lives in so there are no checks to be garnished, no income tax refunds to be held back.
Anyway, that's how child support works in Missouri, and I don't imagine it would be very different state to state.