These are commonly referred to as our Apostles who have a "Special Witness" or testimony of Jesus Christ. 
 
Here is a good example of this: http://lds.org/ensign/2001/04/special-witnesses-of-christ?lang=eng

The Apostles and Seventies are Special witnesses of Jesus Christ, this means that they are called by Heavenly Father to pronounce the existence of Jesus Christ to the world.

But again, aren't we all called to witness to unsaved people that Jesus lives? I'm really having trouble with the designation of special witness. 
 
Linda

http://lds.org/liahona/1995/04/special-witnesses 
 
"The First Presidency and the Quorum of Twelve Apostles are special witnesses who know by personal revelation that Jesus is the Christ. They all have a sacred responsibility to bear their testimonies to the whole world. We sustain all of these Brethren as prophets, seers, and revelators." 
 
When these men testify of Jesus Christ, they are not just proclaiming what they believe to be true. We understand that these men have received personal revelation, beyond what most of us typically receive, that Jesus is the Christ. I would venture that many, if not all of them, have actually seen Christ face to face and spoken with him although they will probably never talk about this in a public setting due to the sacredness of that experience. They *know* that Christ is the Savior. This is what makes their testimony so powerful and, well, "special".