In mythology, damnation was to be condemned by God (or other superior form) to suffer eternal punishment in hell (or something of the equivalence). Any individual subjected to such conditions were known as the damned.
When Kira Nerys learned of the plight of the Ennis and the Nol-Ennis, she feared such a fate as her captors, the Ennis, stating, "We're trapped on this moon with only his forces between us and damnation. We have to defend ourselves to stay alive." (DS9: "Battle Lines")
In 2371, when Neelix explained to The Doctor, that Captain Kathryn Janeway was not aboard USS Voyager, and that she was missing, The Doctor stated that "It seems I've found myself on the Voyage of the Damned. Very well. Please advise the highest ranking officer who is not missing, to see me at his earliest convenience." (VOY: "Time and Again")
Because B'Elanna Torres did not believe in Sto-vo-kor, it was determined that this was cause to bring damnation upon her mother, Miral. When B'Elanna later told her of the efforts she was making to save her, Miral told her that it was essentially too little, too late, explaining that "I would rather face damnation with what little honor you have left me, than cheat my way into Sto-Vo-Kor." (VOY: "Barge of the Dead")