"Direct Contempt of Court" is the inherent power judicial officers possess to maintain respect, dignity, and order during proceedings. Judicial officers are not only Circuit Judges or Federal District Judges, but also may often be specially appointed commissioners or special masters.
A judge may find anyone in her court--attorneys, parties, witnesses, and spectators--in civil or criminal direct contempt. Beacuse direct contempt of court involves conduct at the proceedings, criminal direct contempt is much more unusual than civil direct contempt.
The judge declares that he "finds" the person in contempt, in the sense that the judge is making a finding of direct contempt.
The finding is meaningless, until the judge adds a punishment term. The punishment is mainly a fine or confinement in jail for a brief period of time. Confinement is usually a day or two, but very rarely can be six months or more.
The punishment cannot be completely arbitrary. The judge must make a record about the conduct being punished, and the conduct must either be offensive or interfere with the proceeding. Conduct that shows direct disrespect for the court or the judge is sufficiently offensive.
punishment is immediate.
This is a very rare situation in the U.S. legal system. The contemnor has no right to testify, to call witnesses in his own behalf, to an attorney, to cross-examine witnesses, to a trial by a jury instead of the judge, or to appeal the judge's ruling. Wise judges may grant some of these privileges in the right circumstances, but the judges are not required to do so.
Be careful to distinguish this from indirect contempt of court. A citation for an indirect contempt is very different.
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.