As a non-resident of NY, you will be taxed on any NY-source income. Income is "sourced" where the work is actually (physically) performed. Therefore any income you earn by working remotely from NY is taxable by NYS. It would also be taxable by your resident state of CA. You would therefore file a non-resident NYS tax return in addition your normal CA resident tax return. You would be able to take a credit on your CA return for tax paid to NYS on the income that is taxed by both states, so you wouldn't be double-taxed. Non-residents are not liable for NYC income tax.
NYS and NYC will both tax you as a resident if you maintain a permanent place of abode (such as a leased apartment) in NYS/NYC for more than 11 months of the year and spend 184 days or more (any part of a day is a day for this purpose) in NYS/NYC. If you do this, plus maintain your permanent home or domicile in CA, you would have to file as a resident in both states. In that situation you would be a domiciliary resident of CA and a statutory resident of NYS/NYC - a circumstance known as dual residency.
Here is NY's detailed definition of "permanent place of abode":
https://www.tax.ny.gov/pubs_and_bulls/tg_bulletins/pit/permanent_place_of_abode.htm
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
