The fundamental purpose of insurance is to spread out the risk of individual investments among many parties to reduce the risk to any individual member of the pool in the event that an investment fails. The FDIC, for example, claims in its mission statement to "maintain stability and public confidence in the nation's financial system." It does this by accepting the risk of depositors, which guarantees the solvency of banks.
The forms of insurance most familiar to most people are health, property and life insurance. The purpose behind each of these is similar, and they can be understood on the same terms. Concerning car insurance, About.com brings up the sense of frustration that people sometimes feel from paying their insurance premiums month after month with nothing to show for it. This is understandable, but by grouping many people who seldom or never file a claim together into a risk pool and charging them all a premium, the insurance provider accumulates enough money to pay out to whichever members of the pool suddenly do need help. In effect, the other drivers in the risk pool have covered the inevitable cost of replacing a stranger's car in exchange for the promise that the stranger would do the same for them.