What determines if the person is at fault?

The answer varies, as it depends on the type of Personal Injury case. The plaintiff has the burden of proof in any PI case, to establish all the elements of their claim.
The elements would at a minimum include duty, breach, causation, and damages. The standard of proof in any civil case (including PI cases) is that the plaintiff must demonstrate all the elements by a Preponderance of the Evidence (this is to prove something is more likely than not).