'Civil' cases are brought by individual persons (called plaintiffs) to enforce their private legal rights against other individuals (called defendants). An example is when one party to a contract brings an action for breach of contract against another contracting party.
'Criminal' cases are conducted by representatives of the state (prosecutors) to enforce the provisions of the criminal law that may have been breached by an individual person (the accused). An example is when the state prosecutes an accused person for the crime of theft.