As of August, 2010, every state in the U.S., including Washington, D.C., but excluding New Hampshire, has some sort of law requiring that motorists must prove their financial responsibility in order to drive legally. In these cases, "financial responsibility" really means that drivers can prove they have the ability to absorb a reasonable amount of the medical and repair costs to injured parties in a car accident when they are at fault. (Generally speaking, state governments don't care if you have enough money to replace your own car, or pay your own medical bills.)
For most drivers, "proof of financial responsibility" means auto insurance. All of the states with mandatory insurance laws require some amount of liability insurance, but some also require personal injury protection (PIP) insurance , as well. New Jersey is one such state. As well, many states require a certain amount of "uninsured motorist" insurance, which pays when the person who caused an accident doesn't carry insurance (or enough insurance) of their own.
Many drivers carry more than the required minimum amount of insurance, either because they own homes or businesses and want to protect those assets, or because they have new cars that are leased or financed, and full coverage (liability plus collision and comprehensive coverage) is required by the lender. (Even in New Hampshire, lenders can require that financed or leased cars be insured.)
In many states, proof of insurance is necessary just to renew a driver's license or get state inspection tags, and while it's unlikely that random insurance checks will ever be a part of normal law enforcement proceedings, there are significant fines levied whenever a driver is pulled over for something else (speeding, for example) and is unable to present their insurance information.
But why is it mandatory?
If you consider the sorts of people who don't carry auto insurance, it makes sense that most states have made it mandatory. Uninsured motorists are generally so because they can't afford to pay premiums, let alone the cost of an accident. Alternatively, they may have caused so many accidents that regular coverage is unavailable, and they may be unable to pay for state -sponsored "high risk pool" insurance, or are driving with suspended licenses. There are other reasons people forego coverage as well (if a car is really old, and they don't drive much, for example), but people driving without insurance coverage has become an expensive enough risk that mandatory minimum insurance laws were enacted.
Liability Insurance Protects You by Paying Them
It may help put things into perspective if you remember that mandatory liability insurance doesn't pay the insured driver or their vehicle. Instead, it is purchased to cover the damage claims or medical claims from another person because of an accident. If you are in an accident, and you are determined to be at fault, your liability insurance pays the other driver.
Even though you, as the insured driver, are not receiving money from your own liability coverage, it is a form of protection for you, because it means that your insurance company will support you, either by helping you make your case if you dispute a claim, or by paying claims made by other drivers when you cause an accident. Having that insurance means that you will not have to mortgage your home, sell your business, or take a second job in order to pay for repairs to another person's car.
In a way, it is the very fact that mandatory insurance coverage pays the "other guy" that makes it less controversial than, say, mandatory health insurance. The former is designed to protect others when you are at fault, while the latter is meant to protect you.
Not all states have the same rules!
While almost every part of the United States has some kind of mandatory auto insurance law, the required coverage amounts vary from state to state. If you are moving from one state to another, be certain to check with an insurance agent who represents your new state. Likewise, if you are renewing an auto insurance policy, check with your insurer to see if your state has changed its requirements since your last renewal.