Who is liable and whose insurance covers the accident and any injuries?

By law, the driver's insurance is primary. However, the insurance industry practice is to let the owner's insurance company handle.

The driver and owner of the vehicle responsible for the accident is liable. You don't say who was at fault for the accident, but the nature of your question implies that your boyfriend caused the accident. If so, your boyfriend, as the driver, and your father, as the registered owner, are responsible for the damages caused by the accident. Your father's liability, as owner, is limited by statute to $15,000.00. Your boyfriend has no such limits. The insurance company which insures your vehicle would be the company you advise of the accident and allow it to handle the situation. If it feels that your boyfriend's insurance company should get into the fray, then it will contact your boyfriend's insurance company.

In California, generally insurance follows the car so whichever insurance covered the car is responsible, unless the driver was not a "permissive use" driver.