who is responsible when an accident occurs when there is no insurance in an unregistered vehicle, the driver or the owner?

The driver will be responsible for his actions in causing the accident (subject to whatever defenses he may have). In a few states (I believe NY is one), the owner can be held liable merely by virtue of ownership of the vehicle, but in most states you have to show that the owner is the employer/master/principal of the driver, that the owner negligently entrusted the vehicle to the driver or that the owner provided the vehicle to the driver for a family purpose (Georgia recognizes a "family purpose doctrine" theory of liability against the owner) or similar type theories of owner liability involving agency or active negligence.

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I see that you are writing your question from Bradenton, Florida. Assuming this collision occurred in Florida, the answer to your question is that both the driver and the owner will be held liable for the negligence of the driver. Florida has a doctrine call the "dangerous instrumentality rule" which makes the owner of a vehicle equally responsible with the driver.

In Florida, both the drive and the owner have financial responsibility for negligent operation of a motor vehicle. If both are truly uninsured, you may have possibility of recovery through your own uninsured motorist coverage, presuming you have the coverage. How bad are your injuries?