You buy car insurance to pay for damage you cause to others. Others buy it for when they cause damage to you. When someone crashes into you and its their fault, it was once typical that you would wrestle with the at fault party's insurance company on your own. It was normal since your own insurance company wasn't directly involved.
While the contracts for insurance have not really changed, this practice certainly has, and its only for the better for the typical consumer. Gone are the days when your own company is not involved unless you are at fault.
What typically happens now is each party involved in an accident contacts their own insurance company directly. The individual companies will often handle the claims adjustment and payment themselves, and settle up between themselves as well. It makes for a much more service-oriented and consumer-friendly process.
What typically (but not always) happens with respect to repairing your vehicle is as follows. Lets assume a fender-bender accident with no injuries:
Strictly speaking, as far as you the consumer are concerned the process is the same whether you deal with one company or the other. In actual practice it is far easier for you to deal with your own company, who has a vested interest in providing you with good service, than it is ever going to be dealing with another company you have no ties to. Also, you can pretty well expect that when it comes to properly fixing your car, an insurance company is going to be far better at beating the money out of another insurance company than you are. And they'll do it for free.
This is all a big win for the consumer, but unfortunately it can't all be roses. What if you are injured? well, up to a point you can still expect your own insurance company to handle the matter. Typically they can do this up to the limit of your Medical Payments or Personal Injury Protection coverage. If however your injuries are more extensive, then your own company - who, lets not forget, is not necessarily obligated to pay for any of this - will likely hand off to the other driver's carrier to pay out of the other driver's Bodily Injury Liability coverage.
But if its a simple accident with vehicle damage only, chances are you can make your life easier during a difficult time by simply reporting your accident to your own insurance company and dealing with them directly.