The Assigned Risk Pool
Insurance companies look for customers who are safe drivers and have a low risk of filing claims, avoiding applicants from groups that are riskier. These groups might include young or inexperienced drivers, those with high performance cars, or areas with a high risk of theft or vandalism.
If you can't find insurance, states have set up Assigned Risk Pools where all insurance companies accept policies assigned to it based on the proportion of business that it does in the state. The cost is much higher in the pool than if you purchase the policy directly, but all applicants are accepted regardless of risk.
There are also companies that specialize in higher risk and non-standard policies. Pricing is higher than standard policies, but generally lower than the pool.