A Special Encounter is a type of Card that can be used while playing Eldritch Horror. They are tailored to specific Ancient Ones and will only be used when the Mystery that specifies them is in play. Some Ancient Ones have a Final Mystery that is a Special Encounter. Other Ancient Ones don't have them at all.
Special Encounters are similar to Other World Encounters or Expedition Encounters, in which there is usually a skill check, followed by branching paths that are taken whether or not you pass or fail this check, and usually have differing checks or effects happen to the player. In general, passing the first test will garner some sort of positive effect on the character, while failing will usually become a hindrance. If the Final Mystery is a Special Encounter, some fail effects can instantly devour an Investigator.