It depends on where you live. If you live in a state with “no-fault” insurance, your insurance company covers your bills - to a point. If your medical bills exceed the amount your policy covers, your injuries may be considered "serious" or "significant" and you can pursue compensation outside the no-fault system through a lawsuit. In such a case, the other driver may have to pay you out of his or her own pocket, if you take them to court and win the case.
On the other hand, if you live in a state with “fault” insurance, things are more complicated. Since the other driver has no insurance for you to pursue compensation from, you can try and recover compensation directly from them in a lawsuit. However, if the other driver has few personal assets he or she may not be able to pay you what the court orders. Instead, some insurance policies include uninsured motorist coverage, allowing you to make a claim when the at-fault driver is uninsured.