John the Baptist was the cousin of Jesus and born six months before him. In announcing the impending birth of John the Baptist to his father Zacharias, notice what the angel of the Lord told him in Luke 1:17, "And he shall go before him in the spirit and power of Elias, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just; to make ready a people prepared for the Lord." "Elias" is the Greek transliteration for "Elijah" of the Old Testament. As a matter of fact, Malachi had prophesied that the Messiah would be introduced by Elijah.

Malachi 3:1 Behold, I will send my messenger, and he shall prepare the way before me: and the Lord, whom ye seek, shall suddenly come to his temple, even the messenger of the covenant, whom ye delight in: behold, he shall come, saith the LORD of hosts.
Malachi 4:5 Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD:
Malachi 4:6 And he shall turn the heart of the fathers to the children, and the heart of the children to their fathers, lest I come and smite the earth with a curse.

As part of the long-awaited Messiah package, the Jewish Pharisees and scribes were correctly looking for a predecessor who would introduce the Messiah. It would appear from Luke 1:17 that John the Baptist would be that messenger. And introduce Jesus he did in Matthew 3:11, "I indeed baptize you with water unto repentance: but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire:" As a matter of fact, we see John baptizing Jesus in Matthew 3 and Luke 3.

So, here's the question: Was John the Baptist the fulfillment of Malachi's prophecies concerning the predecessor to the Messiah? The answer may seem strange to you: He could have been, but he wasn't.

First of all, let's look at Matthew's testimony regarding the identity of John the Baptist:

Matthew 3:1 In those days came John the Baptist, preaching in the wilderness of Judaea,
Matthew 3:2 And saying, Repent ye: for the kingdom of heaven is at hand.
Matthew 3:3 For this is he that was spoken of by the prophet Esaias, saying, The voice of one crying in the wilderness, Prepare ye the way of the Lord, make his paths straight.

That's a quotation from Isaiah 40:3 regarding the coming of the Messiah. Luke gives the same testimony in Luke 3:4. An interesting dialogue between John the Baptist and representatives of the Pharisees takes place in John 1:

John 1:21 And they asked him, What then? Art thou Elias? And he saith, I am not. Art thou that prophet? And he answered, No.
John 1:22 Then said they unto him, Who art thou? that we may give an answer to them that sent us. What sayest thou of thyself?
John 1:23 He said, I am the voice of one crying in the wilderness, Make straight the way of the Lord, as said the prophet Esaias.
John 1:24 And they which were sent were of the Pharisees.
John 1:25 And they asked him, and said unto him, Why baptizest thou then, if thou be not that Christ, nor Elias, neither that prophet?
John 1:26 John answered them, saying, I baptize with water: but there standeth one among you, whom ye know not;
John 1:27 He it is, who coming after me is preferred before me, whose shoe’s latchet I am not worthy to unloose.

It appears from this passage that the Pharisees differentiated between Elijah of Malachi 4:5-6 and the Messenger of Malachi 3:1 as seen in verses 21 and 25. It is obvious to us in retrospect that both passages speak of the same individual. So, was John the Baptist Elijah or not? It would appear that "Elijah" is used in the context of addressing a role here in this passage. In fact, John the Baptist was "John," the son of Zacharias, not "Elijah." However, he is introducing the Messiah as was prophesied in Malachi. It is important for the Bible student not to introduce extra-scriptural error into this discussion of John the Baptist being Elijah. For example, if one maintains that John the Baptist was literally Elijah, one has inadvertently introduced the erroneous doctrine of reincarnation into the discussion. John was the son of Zacharias; his soul was not a rebirth of Elijah's soul. Likewise, John's DNA didn't match that of Elijah. In this situation, Elijah wasn't resurrected from the dead; John the Baptist was born of a woman. Therefore, the only way to understand the John the Baptist/Elijah connection is to view John the Baptist as a potential candidate to fill the role of Elijah. However, in order for that role to have been completely fulfilled, the Jews would have needed to accept Jesus as the Messiah at that time followed by the establishment of the Kingdom of God on earth aka the Davidic Kingdom.

When asked about the identity of John the Baptist, here's what Jesus said about him:

Matthew 11:13 For all the prophets and the law prophesied until John.
Matthew 11:14 And if ye will receive it, this is Elias, which was for to come.

Later, however, we see a shift in the presentation of the identity of John the Baptist:

Matthew 17:10 And his disciples asked him, saying, Why then say the scribes that Elias must first come?
Matthew 17:11 And Jesus answered and said unto them, Elias truly shall first come, and restore all things.
Matthew 17:12 But I say unto you, That Elias is come already, and they knew him not, but have done unto him whatsoever they listed. Likewise shall also the Son of man suffer of them.
Matthew 17:13 Then the disciples understood that he spake unto them of John the Baptist.

Jesus sums it up in Matthew 21:32 "For John came unto you in the way of righteousness, and ye believed him not: but the publicans and the harlots believed him: and ye, when ye had seen it, repented not afterward, that ye might believe him."

So, here's the answer you're looking for: If the Jews had received Jesus as the Messiah, then John the Baptist would have fulfilled the role of the prophet of Malachi 3:1 and Malachi 4:5-6. However, since the Jews rejected Jesus as Messiah, then John the Baptist did not fulfill the stated role of Elijah. In other words, John the Baptist could have been Elijah, but wasn't because of the rejection of Jesus as Messiah. Incidentally, Isaiah had prophesied that they would reject the Messiah in his first advent.

As it turns out, it is apparent (though not specifically stated) that Elijah is one of the two witnesses in Revelation 11:3 just prior to the second coming of Jesus. Moreover, Malachi 4:5 refers to "the coming of the great and dreadful day of the LORD:" That is obviously a reference to the second coming of Jesus Christ at the Battle of Armageddon in Revelation 19:11-21.