"No-fault" insurance refers to medical coverage which you are required by state law to carry on your automobile insurance.
Not all states have "no-fault" statutes, though almost all insurance companies sell some type of medical coverage for their auto policies.
Basically, if you have an accident for which you aren't at-fault, and you live in a "no-fault" state, your own insurance must pay for your medical bills. The "no-fault" part comes from the fact that even though someone, say, plowed into the rear of your car while you were stopped at a red light, your own carrier must pick up the ambulance, hospital, rehabilitation, etc.
Some states allow "no-fault" insurance carriers to go after the at-fault party, but this varies too much to discuss here and it's also relatively rare. Typically it's based on the amount of the medical bills or the weight of the at-fault party's vehicle.
Many people who live in "no-fault" states often believe they can prohibit their carrier from paying their bills (with the assumption that they don't want any payments made under their policy in case their rates go up). This isn't the case. Much like worker's comp, "no-fault" medical coverage is primary, and the first-party insurance carrier must pay it.
Finally, many people who know they live in a "no-fault" state often believe this has something to do with physical damage to a vehicle and liability. That's not true. "No-fault" relates only to the medical coverage. If someone hits your vehicle, and he's at-fault, he is still legally liable to pay for the damages to your vehicle.
- It is not necessarily true that in a no-fault state, an at fault party would be responsible for paying for your vehicle damage. Here in Michigan, your own insurance company might file a "mini-tort" with the at fault party's insurance company to recover your deductible, but that's where it ends. Collision and comp coverage is still paid through your own policy.