Withholding is tax collected at the source: your employer takes it out of each paycheck before paying you and passes it to the tax authority. It is a prepayment, not the tax itself. Your actual liability is settled for the whole year, and what was withheld is credited against it, giving you a refund if too much came off or a bill if too little did.
In the United States the employer works out federal income tax withholding from the Form W-4 you hand in, which tells payroll your filing status and any adjustments you want made, according to the IRS. The IRS suggests filing a new one each year and whenever your circumstances change, and its Tax Withholding Estimator can produce a filled-in form. Social Security and Medicare are withheld too, at fixed rates set by IRS Topic 751 rather than by anything on the W-4.
A big refund is not a windfall. It means you lent the government money interest free during the year, and reducing your withholding would have put it in your pay instead. Owing a large amount at filing time can bring a penalty, so the aim is a withheld total close to the final bill.
Other countries run the same idea under their own names. The UK, Ireland and New Zealand call it PAYE, and there payroll works year to date against your tax code, which keeps the running total close to the final figure. Elsewhere a year end return reconciles the two.
The take-home pay calculator works out the tax owed for the year and spreads it across pay periods, which is what accurate withholding would take. Your payslip will differ if your W-4 asks for more or less than that.