“Indirect contempt of court” is the violation of a court order outside the immediate presence of the court. It may be either civil, or criminal.
A charge of "Criminal Indirect Contempt of Court" is a criminal charge. Defendants charged in this manner are afforded all the usual privileges of other criminal defendants, like the right to a trial by jury, to examine, and to call, witnesses, and to testify on one's own behalf.
In contrast, civil contempt is not a criminal charge. Civil contempt of court is used to coerce a party to perform an action. When discussing civil contempt, commentators often use the phrase "the contemnor has the key to the cell in his own pocket". What this means is that the contemnor will be punished for only so long as they refuse to perform the action required of them by the court. Once the contemnor performs that action, they will be immediately released from whatever sanction the court has imposed.
For a discussion of the various kinds of contempt, but primarily direct contempt, see Parkhurst v. U.S. Dept. of Education. The dissent in Parkhurst also makes interesting points.