You will need to file in this order:

• MD non-resident
• DE resident

Always file the non-resident state first.

You must file in both states.  You'll be able to take a credit on your DE return for the taxes paid to MD, so you won't be double-taxed.

Your employer is correct to withhold MD taxes, since most, if not all, of your total state tax bill will be to MD.

Thanks Tom. Which state return should I complete first?

As Joseph and Tom noted, you should do your nonresident state (Maryland) first so that the Maryland tax can be calculated, and then your state of residence (Delaware) so that the Maryland tax paid can be applied as a credit on the Delaware return.

Thanks, Joseph.

Thanks Bill, and as well again to Joseph and Tom. Unsure as to why I hadn't seen their answers until yesterday.

My understanding is that, if you are a resident of Delaware and you earn income in MD, you don't owe taxes to MD. If taxes from MD employment are withheld, then you must file to have them refunded. My question is whether I can ask my MD employer not to withhold taxes so that I do not have to file a MD return. 

@stevef 

"My understanding is that, if you are a resident of Delaware and you earn income in MD, you don't owe taxes to MD"

No, that's incorrect.  DE residents who work in MD are subject to MD state income tax on their MD-source income.  MD has tax reciprocity with all its bordering states EXCEPT Delaware.  As a DE resident, your income is also subject to taxation by DE.  Therefore you must file a tax return in both states.   Your employer is correct in withholding MD taxes.

You will be able to take a credit on your DE return for taxes paid to MD, so you won't be double-taxed.