Federal income tax on your paycheck is based on an estimate that you're responsible to calculate when you file the w-4 when you start working at a new employer (it can be amended at any time). Those paycheck estimated payments are added up and deducted from what you actually owe at the end of the year and result in either a refund (you overestimated) or taxes due (you underestimated).

If your net federal income tax is $0 there is an exemption box on the w-4 you can check. That causes no federal income tax to be withheld, but the same end of year calculation may find you owe money if you made more than expected.