Ordinarily, you are required to file a state income tax return in every state where you earned an income.  There are some exceptions, such as military personnel who are deployed outside of their state of record.  Some states will let you slide if you only earned a very small income while you were there, and some states have no income tax whatsoever.

Members of the military services designate their home of record as the state in which they enlisted.  Military personnel are usually considered to be residents of their state of record.  The Federal Government prohibits other states from taxing the wages of non-resident military personnel stationed there.

If you made a permanent move from one state to another, you are considered a part-year resident of each state.

If you traveled from your home state to take on work in another state and maintained a permanent residence in the state from which you traveled, you are a non-resident of the state in which you were working.

When considering whether your move from one state to another is permanent, tax wise, everything goes to intent.  If you intended to take up a permanent residence in a state, then you were a part-year resident.  If you were traveling from assignment to assignment and remained in one state until that assignment expired, and then moved on to the next state, you were likely a non-resident.

One of the easiest ways to determine how much money you owe each state in the course of a year of travel is to let a tax software program figure it out.  A tax software program will take all of the information that you provide on your federal income tax return, including residency status, and transpose it onto the correct lines in a corresponding state tax form.  The software program will even help you decide what income is taxable and what deductions you receive in each state where you earned income.  It will also tell you what filing requirements must be met in each state where you had earnings.