Over the coarse of the show there have been mention and brief showing of some major characters, people or in some cases things that seem to be able to do things without any explanation. They seem to be able to appear or disappear at will, cure illnesses or control the outcomes of the islands inhabitants. The show has yet to explain these strange phenomena's but as the show comes to its last season more and more questions are being answered.
*WARNING - THE FOLLOWING INFORMATION MAY CONTAIN SPOILERS IF YOU ARE NOT CAUGHT UP IN THE SHOW.
Jacob - Jacob is the true leader of the others, the current leaders are supposed to come to him for guidance and then pass the information back to the others. Jacob lives in a cabin which does not have a fixed location on the island, but rather you can only approach it if he summons you or you know where it will be. When Ben was in charge he was unable to see Jacob which he thought meant he was unworthy, therefore he just made up orders and issued them himself under Jacobs name. The reason he couldn't see Jacob however was because Jacob had abandoned his cabin long ago for fear of being discovered by his enemy, the bearded man.
The Bearded Man - The Bearded man was introduced in a flashback at the end of the fifth season as an antagonist to Jacob. The flashback went back to the time of the first inhabitants arriving on the island suggesting that neither him nor Jacob ever age. The bearded man expressed distain for allowing people to come to the island but Jacob thought they need people there. A reference was made to a deal that is in place between the 2 of them where neither one is allowed to kill the other. In the present the Bearded Man appears as John Locke to Ben and tricks him into killing Jacob for him.
Christian Shepard - Jack and Claire's father, he died in Sydney before the plane crash and Jack was flying his body home for a burial. Even though he died before the plane crash he is still seen walking around the island by almost all of the characters at some point and has talked to some of them. After Jacob left his cabin Christine took up residence there which leads fans to believe that he is not on Jacobs side, possibly on the side of the bearded man but neither of their motives are yet known. In addition to his strange on the island appearances he has also appeared in over half of the characters flashbacks at some point and seems to be the biggest correlation between most of them.
Smoke Monster - The smoke monster was introduced in the pilot episode of the show but was not revealed until the season finale. It appears as a column of black smoke at seemingly random times and attacks the castaways. It has also appeared many times in the form of people that the castaways know. Since it can take many forms their is speculation as to whether it may also be Christian, the bearded man or both. The others call it a security system which guards a certain part of the island. It is unable to travel onto the beach or around banyan trees which have the strongest roots so it is theorized by the fans that the smoke monster is someone guided by underground tunnels which couldn't go under the beach of through the thick roots. The writers have stated that by the end of the show the smoke monster will be explained.