Quasi-criminal is a type of case that is based in civil or common law, but which has elements that can overlap with criminal law as well. An example of this is a family law case in which there is the possibility that a court order was not kept; the judge in the case could order jail time for the person that breached the order, making something that was once a civil matter have elements of criminal law as well.
Contempt of court is another example of something in law that is quasi-criminal. If a person in a civil law case refuses to give evidence after they were subpoenaed as a hostile witness, the judge could order jail time. By virtue of the contempt of court resulting in jail time, the case would be considered quasi-criminal, as it happened within the context of a civil law case.