The issue you might be running up against here is that the payroll person is doing a poor job of explaining what withholding tables are and how they work. Basically, when you fill out a w-4 form and tell your employer what your marital status is and how many dependents you will be claiming on your tax return, you are actually giving them information for the employer to project your total tax amount at the end of the year. So if you have 4 children and are filing as head of household (for example), that tells the employer that the first $19,400 of income you make will have no income taxes owed, and that additional amounts you make above that amount will be taxed according to the tax table for the filing status of head of household. In addition, you are telling them that you will have child tax credits that will offset your remaining taxes (how much tax will be calculated on your income - your refund or amount owed at the end of the year is based on this number.
The purpose of income tax withholding is not to give you a big refund, but to cover your actual taxes. So if you claim a lot of dependent children, the estimate will be that you don't owe any tax - and thus, no taxes are withheld for federal income taxes. This almost always happens with part-time jobs, because the projected income of a part-time job is usually too low to project (estimate) a tax amount owed (and thus trigger income tax withholding).
If you would like your employer to withhold more taxes than they are, you can request "Extra Withholding" on your w-4 form on Line 4(c). You can also claim less children on your w-4 form - the w-4 form is only to gather information for the employer; it is not reported to the federal government and does not have to match your tax return. I have seen employees claim they have 19 children to avoid having any income tax withheld from their pay - I don't recommend this, but I have seen it done. The easiest way to trigger normal tax withholding is to claim yourself as single with no dependents. This might be what the payroll person is trying to say - that she is bound by the withholding tables that are based on the w-4 information you provided. If you update this information, you can change the amount of federal income tax that is withheld from your paycheck.
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