It depends on which states.
The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state.
Many neighboring states have reciprocal agreements and it is not necessary to file a work state return.
When you worked in a state without an income tax (e.g. Texas or Florida), there will be no credit, since there was no TX/FL tax. In other words, having worked in a state without an income tax does not get you out of paying state tax on that income, to your home state.
