The majority of personal injury cases involve negligence--e.g., a doctor failing to properly diagnose a patient, or a driver falling asleep at the wheel--which results in pain and suffering for the plaintiff. However, some types of personal injury are inflicted intentionally.
However, with intentional torts--offenses such as assault, battery and other types of deliberate conduct that cause harm to another person--the defendant could have two cases on their hands at once: one civil and one criminal.
For example, let's say you and your friends go out drinking. At the bar, another patron makes a degrading comment toward you, and you punch them in the face--shattering their jaw. The manager calls the police, and you're arrested.
If the state chooses to press charges against you, then you have a criminal case to contend with. With criminal charges, the case is between you and the government, and you're looking at the possibility of jail or prison time.
In addition, the man with the broken jaw could also sue you for his medical expenses and resulting lost income from work. This is an example of a civil case, which is between you and the injured party. In such cases, the penalty is typically owing money in damages.
It is worth noting that being tried in civil court and criminal court for the same offense does not constitute double jeopardy--your Fifth Amendment protection from being tried twice for the same crime--because this rule only prevents having more than one criminal case tried against you for the same crime.