What happens if the driver who hit me was uninsured or under insured e.g. had insurance but the insurance is less than my case is worth?

If the driver that hit you was uninsured or underinsured check your policy to see if you have this type of coverage.  If there is no coverage anywhere its probably a case that has no value.  To be absolutely sure consult with a personal injury lawyer and make sure the consultation is free.  The exception to this rule is a defendant with assets.  If someone with means hits you and does not have insurance a personal injury lawyer may be able to make a recovery.