What things did John the Baptist say that the elders and priests did not believe? In Matthew 21:25 we see priests and elders reasoning within themselves:


  The baptism of John, whence was it? from heaven, or of men? And they
  reasoned with themselves, saying, If we shall say, From heaven; he
  will say unto us, Why did ye not then believe him?


My question here is how did they not believe him? What exactly did they do that showed that they did not believe him? What things exactly (of what John the baptist was saying) did they not believe? Was it something like they had refused to be baptized from John? 

(Answers supported by the verses from the Gospel of Matthew are most preferable)
Well, in the preceding verses, they were questioning Jesus about his authority.  Jesus answered by asking them about John the Baptist's authority, to draw an analogy.  So the obvious answer would be that they didn't believe John's testimony of Jesus:


  John 1:19-34
  
  19 And this is the record of John, when the Jews sent priests and
  Levites from Jerusalem to ask him, Who art thou?
  
  20 And he confessed, and denied not; but confessed, I am not the
  Christ.
  
  21 And they asked him, What then? Art thou Elias? And he saith, I am
  not. Art thou that prophet? And he answered, No.
  
  22 Then said they unto him, Who art thou? that we may give an answer
  to them that sent us. What sayest thou of thyself?
  
  23 He said, I am the voice of one crying in the wilderness, Make
  straight the way of the Lord, as said the prophet Esaias.
  
  24 And they which were sent were of the Pharisees.
  
  25 And they asked him, and said unto him, Why baptizest thou then, if
  thou be not that Christ, nor Elias, neither that prophet?
  
  26 John answered them, saying, I baptize with water: but there
  standeth one among you, whom ye know not;
  
  27 He it is, who coming after me is preferred before me, whose shoe's
  latchet I am not worthy to unloose.
  
  28 These things were done in Bethabara beyond Jordan, where John was
  baptizing.
  
  29 The next day John seeth Jesus coming unto him, and saith, Behold
  the Lamb of God, which taketh away the sin of the world.
  
  30 This is he of whom I said, After me cometh a man which is preferred
  before me: for he was before me.
  
  31 And I knew him not: but that he should be made manifest to Israel,
  therefore am I come baptizing with water.
  
  32 And John bare record, saying, I saw the Spirit descending from
  heaven like a dove, and it abode upon him.
  
  33 And I knew him not: but he that sent me to baptize with water, the
  same said unto me, Upon whom thou shalt see the Spirit descending, and
  remaining on him, the same is he which baptizeth with the Holy Ghost.
  
  34 And I saw, and bare record that this is the Son of God.