Registering a NJ LLC but revenue from NY
Being the LLC will be formed in New Jersey, you would have to file as a foreign entity in New York if you plan on having a physical location there, hiring employees in the state, or conducting a consistent amount of business there.
For example, if you have the business registered in NJ, and were selling to NY customers through a website, then you would not need to file a foreign entity in New York. Alternatively, if you were to say, open an office or conduct your business in person, then you would need to file the foreign entity.
Regarding taxes, LLC's are pass-through entities, which means that regardless what state(s) the business is registered in, the income would be passed through directly to the owners' personal income, where you would then pay your personal income taxes via federal IRS and state of residency.
It is always a good idea to consult a licensed accountant when planning to operate your business in multiple states.