How do the federal and state pieces of my income tax fit together?

Federal and state income taxes are separate systems. You file a federal return with the IRS and, in most states, a separate state return with your state's tax agency. Many states start from your federal adjusted gross income and then apply their own deductions, credits, and rates, so your federal numbers flow into the state calculation. A handful of states levy no income tax at all. State rates and rules vary widely; some are flat, some progressive. Because state income tax you pay can be part of your federal itemized deductions (subject to the IRS cap), the two interact. When you change your W-4, remember many states have their own separate withholding form to update as well.