In 2009 I lived in NYC for one month and then I moved to NJ for the remiander of the year. How do I file? I work from home so I worked in NY when I lived in NY and worked in NJ when I lived in NJ.

On NY tax return, report income for one month as part year resident.
On NJ tax return, report income for the remainder of the year.

You must file as a part-year resident of NY (and NYC) and also as a part year resident of NJ. You actually have to report ALL your income to BOTH states, but as you work through the math you'll see that each state taxes you only on the amount you earned while a resident of that state, however using a tax rate that's determined by your total income.