The points system is how many states keep easy track of your driving violations. Where points are used, the driver receives points for many driving violations, with additional points assessed for more severe instances. When the insurance company pulls your driving history, it will list infractions, not simply points. Even though it may appear that your accumulated points are causing your car insurance rates to increase, it is actually the frequency and severity of the infractions, which correspond to the number of points.
In most states, car insurance companies can only reference your driving history for the last 2 to 5 years. This allows a person with a bad driving record to clean it up and eventually even gain a safe driver status. It should be noted, however, that some offenses, including DUI, may stay on your driving record for a much longer period, and some states allow DUI convictions to remain with you permanently. For example, a speeding ticket may expire from your driving record after only a couple of years, but a single DUI conviction could remain for 10 years or more, and have a serious impact on the cost of your insurance.
Another impact that sometimes appears to be point related is that your insurance costs will go up if your license has been suspended or revoked. The actual cause of the increase is that you are often required to carry an SR-22 after serious infractions, and that cost is added into the cost of your car insurance. An SR-22 is not insurance itself, but merely a certificate presented to the DMV by the insurance company which states that you are in compliance with insurance requirements.