Murder is a criminal charge brought by the state in the name of the people of the state. When the family of a deceased person files a wrongful death lawsuit seeking monetary damages against the person they believe is responsible for the death of their loved one, the suit is filed in civil court in the name of the family.
The burden of proof is different in civil and criminal cases. In a criminal case, the defendant is presumed innocent and the state must prove the charges "beyond a reasonable doubt." This is because a person's liberty is at stake. A "not guilty" verdict means the state has not fulfilled its burden. In a civil case, only money damages are at issue, and the burden of proof is the lesser "preponderance of the evidence" standard, which essentially means a finding of 51% or "more likely than not."
Also, in a criminal case, the defendant cannot be forced to testify. The jury is instructed it may not consider the defendant's silence in reaching its verdict. There is no such protection in civil suits. Another difference is that in a criminal case, the jury's verdict must be unanimous. In civil cases, depending on the jurisdiction, the defendant can be found liable if a lesser number of jurors agree on a verdict.
The most prominent example is the case of O.J. Simpson. The state charged him with first-degree murder, and failed to prove the charge beyond a reasonable doubt. A verdict of "not guilty" resulted. The families of Nicole Brown Simpson and Ron Goldman then sued Simpson in civil court, alleging he wrongfully caused their deaths. After being instructed to decide the case using the "preponderance of evidence" standard, the jury found in favor of the Brown family on a battery claim and in favor of the Goldman family on the wrongful death claim. Simpson was not imprisoned as a result of the civil jury's verdict, but ordered to pay $33.5 million in damages. An appeals court upheld the award.