Typically most insurance companies will want you to have insurable interest in the vehicle in order to get insurance on it. Having insurable interest means being an owner of the vehicle, co-signer or lien holder.
Some CarInsurance.com companies require that your name appear on the title or registration in order to be able to purchase an auto policy in your name. There are some insurance carriers, such as Esurance, though that allow you to insure a motor vehicle without having insurable interest in it.
So it may be possible for a car to be owned and registered by one person and insured be in another person's name but it will depend upon state insurance laws and insurance companies' guidelines if it can be done in your state and for your situation.
A car cannot be owned by one person and insured under another person's name though for the purpose of getting cheaper insurance on the vehicle. If you want to insure a car that you do not own you will need to make sure the insurance company understands the situation and permits it. If you instead do not inform the insurance company properly that the car is now owned by the person insuring it and do it only to try and reduce the insurance costs, most states will find this to be misrepresentation, a form of insurance fraud.