State laws can vary, but in general a car only needs to be registered in order to be operated, and there are typically exceptions/grace periods associated with moving and new car purchases. That's the law in New Jersey, but people have been ticketed for having an unregistered vehicle parked in their driveway. It does not need to be registered to sell, the new owner will have to register it in PA.
The buyer in PA will follow these steps, per PA DOT:
- Before you clinch the deal, arrange to have car insurance.
- Make sure the seller completes Section A on the back of title.
- Print and sign your name on the title in the presence of a notary.
- Record the vehicle's odometer reading.
- Visit a PennDOT Driver and Vehicle Services office to transfer the title to your name and register the vehicle. You may apply for new license plates or make arrangements to transfer plates from another vehicle.
If the vehicle's emission and inspection decals are not valid, you will have 10 days to pass the inspections.
If they'll be driving it home, they'll need proof of insurance to get a temporary plate from NJ for $5 at any NJ MVC Agency that will give them 30 days to get it registered in PA.
PA assesses sales tax on the current fair-market value of a used car at time of registration, people historically put $1 as the sale price to skirt sales tax, but states have wised up.
Edit: This article suggests that gifted cars do not incur sales/use tax, so long as you have proof of sales tax paid, but they might mean PA sales tax. And if taking a payment it's not really a gift, one of those many cases where it's tempting and unlikely you'd get caught, but ultimately would be committing tax fraud.