Throughout the Bible, Book of Mormon, Doctrine and Covenants, and Pearl of Great Price there are passages referring to God as "one". 
 
This is interpreted to mean one in purpose, spirit, and unity. Like how married couple is "one", a soccer team is "one", or how a family is "one". Certainly it's doesn't mean they are the same person.  
 
The Godhead (or Trinity) is clearly made up of 3 individuals as evidenced at Jesus' baptism, the mount of transfiguration, the great intercessory prayer offered in Gethsemane, the final moments on the cross, Christ's appearance in the western hemisphere, and God and Jesus' appearance to Joseph Smith. In all of these cases, the father, son and/or holy spirit were evidenced as completely separate beings.

I think it is also important to remember that Jesus (in John ch. 17:21-22) - in praying to His Father (Heavenly Father) - said, 
21 That they all may be one; as thou, Father, [art] in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me. 
 
22 And the glory which thou gavest me I have given them; that they may be one, even as we are one:  
 
To me, I believe that Jesus is showing us what is meant by being one; or should we really think that this means that we will all be just one person?