Specifically, if a committee votes to cite someone for contempt of the committee, a resolution would pass to the full chamber. The full chamber, House or Senate, then may or may not pass it. If the full chamber passes the resolution, there is more than one option with which to enforce it.
Although it has not been used since the early part of the last century, a chamber may on its own send its Sergeant-at-Arms to arrest the individual found in contempt.
To go the route of criminal prosecution, the matter would be referred specifically to the U.S. Attorney for the District of Columbia who has the "duty" to refer the matter to a grand jury. That said, because there are disagreements over executive power (namely the proponents of a "unitary executive theory" arguing that compelling the U.S. Attorney to hold someone in contempts amounts to compelling the President himself to do something which would be a violation of the separation of powers) that "duty" is not always upheld.
Finally, the Senate Legal Counsel may be directed to file a civil action against an individual found in contempt. The federal district court, upon motion by the Senate, would issue another order directing the individual to comply with the Senate. If nothing happens, then the person may be found to be held in contempt of court.