In order to understand the differences between these two, let us discuss each of these offenses.
Civil contempt generally occurs when a person fails to carry out an order that a court has issued. A common example of this comes when someone fails to pay child support as ordered.
Criminal contempt is an offense that hurts the ability of a court to do its work. A common example of this would be when a lawyer violates a gag order that is imposed in the hopes of ensuring that a trial can be conducted in a manner that is fair to both sides.
With this in mind, we can say that civil contempt is usually committed against an individual (like the person who was supposed to get the child support payment). The court usually only uses charges of civil contempt as a way to force the violator to obey the order given. By contrast, criminal contempt is an offense against the whole system of justice and, by extension, against society as a whole. When this happens, courts will use charges of criminal contempt as a way to punish the violator (as opposed to simply trying to get him or her to comply with a court order).