When the time comes to pay the IRS, it doesn't matter which state you live in. If you have taxable income, you owe every penny, regardless of whether your employer took the pennies out of your paycheck. That's the law whether you live in North Carolina, North Dakota or north Arizona. (ref1)
Your employer bases withholding on how much money you make and the number of withholding allowances you claim on form W-4 when you start work. If you made a mistake -- you and your spouse both claimed the same allowance, say -- that can leave you with too little withheld from your check each pay period. It's also possible your employer miscalculated or wrote down the wrong number somewhere. When tax time rolls around, you have to pay the right amount even if it wasn't withheld.
Not everyone gets tax withheld from their pay. With household workers -- cooks, housekeepers, maids, for instance -- it's optional: your employer only takes out money if you request it. That doesn't let you off the hook for paying the tax, however. There are also some kinds of pay that aren't subject to tax. If you leave your job for military duty but your boss still pays you, the pay isn't subject to Social Security tax -- so in that case, that tax wouldn't be withheld and you wouldn't pay it.
If you're successfully self-employed, you may be getting paychecks from multiple clients -- and none of them will withhold taxes. When you work for yourself, there's no boss to take money out of your checks, so you pay estimated tax four times a year instead. When April 15 rolls around, you pay the IRS if your estimate was low, or request a refund if it was high. If you owe estimated tax and don't pay it, the IRS can hit you with fines and penalties.
If you discover your employer didn't withhold the correct amount, ask her to find the error and fix it. If the error was yours -- you got divorced mid-year and didn't adjust your withholding to single, for instance -- update your W-4. Should you discover your employer's not withholding anything, talk to him about it and maybe contact the IRS to alert them. Once you've fixed withholding for future paychecks, hopefully you won't have to write the IRS such a big check for next year's tax bill.