As I said, I was wondering how things were arranged in the US without being judgmental about it. I got my answer, thanks.
Edit: and what happens if the person that injures you has no insurance ?
Kind of depends on the circumstances. If it's an automobile accident, they're in deep shit. In most states in the US, it is
required by law to have a minimum level of liability insurance coverage if you are driving a motor vehicle, for just this sort of reason.
If you are injured through some other sort of act it depends on who is at fault and what the nature of the injuries are. If I'm hurt on the job, my employer is responsible for covering my health care costs associated with the injury. If they don't/can't/won't pay, then they get taken to court, and if I have a good case, they get their asses sued off.
But if it's just some random idiot that walks up to me and hits me with a baseball bat or something, there are other legal recourses.
My point is, in cases such as driving or working, there are requirements with regard to insurance, and a lack of compliance with said requirements does not in ANY way absolve the liable party of responsibility if their actions or negligence results in injury to someone else.