State laws differ but typically you must just have proof of insurance, your insurance card normally, available to be shown upon request by law enforcement, the DMV and other state entities. Also if you were in an accident than you and the other driver should show your insurance cards when trading information.
For example the California Department of Motor Vehicles (DMV) notes that since 2006, all insurance companies are required to report insurance status information to DMV for all private use vehicles (CA Vehicle Code 16058) and law enforcement and court personnel have access to DMV records to verify that your California registered vehicle is currently insured (CVC 16058.1). The CA DMV though specifically and emphatically states that even with these laws in place one must still carry proof of insurance in their vehicle at all times.
The New York Department of Motor Vehicles also notes that one should keep an insurance card with the vehicle. If a police officer requests your proof of insurance, you must show your Insurance Identification Card in NYS.
Even if your specific state's statutes and law do not expressly say that the insurance card must be in a vehicle with a person while they are driving, it is always advisable to carry the car's insurance card with the vehicle in case a law enforcement officer requests proof of insurance or if you must provide this insurance information to others after being in an accident.
As long as you have proof of insurance to show when requested it normally does not matter if you keep it in your wallet or in the car itself. If you loan out your car to others it would make sense to keep the insurance card in your vehicle so that the driver borrowing your car has proof if they need to show it to law enforcement. Also there is the chance that you could leave your house without your wallet or have your wallet stolen and just happen to need to show your insurance and be unable to do so. Failure to show proof of insurance can end up with you paying fines or at least the hassle of going to court to show that you did have insurance in effect on the day of the traffic stop or accident.
Some insurance companies provide you with 2 insurance identification cards for your vehicle. This way you can carry one on you personally, in your wallet, while the other one stays with the vehicle.
So it is wise to keep your insurance identification card in your car. For insurance purposes the card itself should be enough to be kept in the car to provide proof of insurance to law enforcement or to give others information on your insurance company. As for other things to keep in your car, in most states you must also keep proof of registration on the vehicle with you when driving. Police do typically ask for insurance card and registration. In some states though the registration is shown with a sticker on the front windshield.
And there are some special states where proof of insurance is on the registration card. For instance in Massachusetts they do not use insurance cards as proof of insurance. Instead a vehicle's Registration is used as proof of insurance in MA. This is done since when a driver gets auto insurance in MA their insurer will stamp an application for registration as proof that the car owner has purchased the minimum Liability insurance or more.
While what documents you should have in your vehicle vary by state in general you should carry your insurance identification card and your registration. Your actual insurance policy you can keep at home in a safe place normally. You need to also have in your possession your driver's license however you would usually carry this on your person instead of leaving in say the glove compartment of your car with the insurance card and registration information.
If in an accident it is convenient to have also some paper to write down the other person's information or to write your information down and share it with the other party. Also if you do not have a phone that takes pictures than a disposable or other type of camera is good to have on hand if you are in an accident and need to take photos of how the cars ended up, the damage, etc.
You can contact your state's DMV or insurance regulator for them to advise you on your state's laws regarding carrying proof of insurance (an insurance identification card) and what other documents they recommend keeping with you in your vehicle at all times. If you want to shop around discount auto insurance click here.