When a person unlawfully kills someone else they have committed a crime against the state (murder, manslaughter etc.) and a civil wrong (a tort) against the victim (specifically, the tort of wrongful death).
The state responds with charging the perpetrator with a crime and attempting to prove that they did the deed "beyond reasonable doubt". If convicted the criminal is "punished" by incarceration or death or a fine or community service etc.
The victim (through their estate) or other affected people (family, dependents etc.) can respond by suing the perpetrator for compensation (money) for the damage they caused. If the plaintiff proves their case "on the balance of probabilities" the perpetrator will owe the plaintiffs whatever damages the court awards.
There are many crime/tort mirrors (many have the same name): theft and conversion, trespass and trespass, fraud and fraud, kidnapping and false imprisonment etc. Because of the different threshold of proof between criminal and civil liability, it is much easier to win a civil case.