In theory, criminal law deals with wrongs against the state, or against the community as a whole, while civil law deals with wrongs against private individuals.
Although criminal cases often involve an individual who is the victim, they deal with the element of the wrong done which effects the whole community - usually the use of violence, deception, or violation of property rights. Most criminal offences will also involve the commission of a civil wrong - a tort. For example, the offences of battery and GBH would also involve the tort or trespass to the person - for which the victim could sue the defendant in civil law.
The distinction between civil and criminal law has not always been sharp - in the early common law, there was no such distinction, and even as criminal offences began to separate from their civil counterparts, victim were still usually left to persue punishment for the criminal.