If you caught as operating a vehicle as an unlicensed driver in New Jersey, you may be charged with driving without a license.
According to NJ Statute 39:3-10 a person violating this NJ law regarding the need to be licensed to operate a vehicle shall be subject to a fine not exceeding $500 or imprisonment in the county jail for not more than 60 days. The New Jersey law goes on to say that if that person has never been licensed to drive in this State or any other jurisdiction, he shall be subject to a fine of not less than $200 and, in addition, the court shall issue an order to the commission requiring the commission to refuse to issue a license to operate a motor vehicle to the person for a period of not less than 180 days.
So if you are unlicensed and want to obtain a driver license after being convicted for driving without a license the New Jersey Motor Vehicle Commission can refuse to issue a license to operate a motor vehicle to you for a period of 180 days (6 months) after the date of the conviction.
This NJ Statute regarding the need for a driver to be licensed does not state that the penalties change for second offense or conviction for driving without a license so if you were ticketed for this violation you should contact the court listed on the ticket for information on the penalties you will be facing in court.
If your license had previously been suspended and you are found driving you can get up to a $500 fine and up to 6 additional months of your license suspension. This is for a first offense. If it is your second offense the fine is $750 and up to 5 days of jail and up to 6 months additional license suspension.
As for the issue of you being an illegal person, this appears to mean that you are not legally in the United States and thus are an illegal immigrant. This issue would be one that law enforcement in New Jersey would normally refer to the United States Immigration and Customs Enforcement center. This federal agency would then look into your status and likely either file immigration charges against you or perhaps even deport you back to your home country.