Will insurance cover someone other than the insured?

My dad and I live separately. But if I were to borrow this vehicle and get into an accident, would I be covered under his insurance or mine?

While it can vary by company, insurance will follow the car rather than the driver. So, in the event you get into an accident in your dad's vehicle, his insurance would be primary and yours would be secondary. Now, the type of coverage you would receive would depend on what coverage your dad has and the accident in particular.