Usually, the at-fault party or their insurer.
In the case of an accident caused by the negligent driving/use of a motor vehicle, the compulsory third party (CTP) insurer of the at-fault vehicle will be responsible for paying your compensation.
In the case of an accident caused by the negligence of another person (such as a local council), that negligent person will be responsible for paying your compensation.
Thankfully, most at-fault persons have some form of insurance (such as CTP insurance or public liability insurance). That means your claim is very likely to be against an insurer (and not an individual person).