Context of Matthew 16:13-19
The site of ancient Caesarea Philippi is in the northernmost part of Israel at the foot of Mount Hermon. The only Biblical references to the city are found in the New Testament in Matthew 16:13 and Mark 8:27 which describe Jesus’ visit to the vicinity of Caesarea Philippi with his disciples. This trip may have been for leadership training or proselyting or a combination of both. In answer to the Savior’s question on the occasion, Peter bore his testimony on behalf of the rest of the apostles with the fervent declaration “Thou are the Christ, the Son of the living God.” The Master responded by explaining that such a testimony came only through revelation from God, and then declared that his Church was to be built “upon this rock” of the revelation. He promised Peter and the Apostles “the keys of the kingdom of heaven” which would empower them to perform works that would be sealed in heaven as well as on earth. Just a week later these keys were bestowed to Peter and his associates on the Mount of Transfiguration. 
“Thou art the Christ, the Son of the Living God”
Peter’s declaration to the Son of God, that he was indeed the Christ gives us a window of opportunity to discuss Peter’s conversion and testimony. By the power of the Spirit we come to know the things of God—we gain a testimony. By that same power we continue the process of conversion. In a sense, gaining a testimony is a form of conversion from unbelief to belief, from doubt to certitude. Peter’s announcement that Jesus is the Son of God, is one stone building upon another of his conversion and testimony.
The following is a discussion by Robert L. Millet on Peter’s conversion and subsequent testimony.
The importance of building upon one’s testimony and thereafter becoming truly converted is illustrated dramatically in the life of Simon Peter. Peter was a humble man, a fisherman, when he was called by Jesus to the ministry. As a member of the meridian First Presidency, Peter was frequently alone with the Savior and privy to many of the spiritual experiences recorded in the New Testament (Matthew 14:28-29; 17:1-9; 26:37; Mark 5:35-43). There is little doubt that Peter was a good man with sound desires, one who had a testimony of the divinity of Jesus of Nazareth. After Jesus preached his powerful bread of life sermon, in which he identified himself as the true bread of life and the living manna, many of the disciples were offended and “walked no more with him.” Christ turned to the Twelve in what was a poignant moment and asked: “Will ye also go away?” Peter responded in deep sincerity and conviction, speaking for the others of the Twelve: “Lord, to whom shall we go? thou hast the words of eternal life. And we believe and are sure that thou art that Christ, the Son of the living God” (John 6:66-69).
At Caesarea Philippi some six months before the crucifixion, Jesus asked the Twelve: “Whom do men say that I the Son of man am?” Again answering for the Twelve, Simon said: “Thou art the Christ, the Son of the living God” (Matthew 16:13, 19).
Although the New Testament attests that Peter had a testimony, it also affirms that he slipped and stumbled. Not infrequently, he was chastened by his Master for his shortsightedness and impulsiveness. Almost immediately after his remarkable testimony at Caesarea Philippi, Jesus began to prepare his chosen Twelve for what lay ahead—his arrest, passion, and death. “Then Peter took him, and began to rebuke him, saying, Be it far from thee, Lord: this shall not be unto thee. But [Jesus] turned, and said unto Peter, Get thee behind me, Satan: thou art an offense unto me; for thou savorest not the things that be of God, but those that be of men” (Matthew 16:21-23). Here we see that the rock, or seer stone (JST, John 1:42), had become a stumbling block. And, of course, there was the most classic of Peter’s blunders—his open denial of a knowledge of Jesus on the night the Savior was arrested (Matthew 26:69-74).
How could someone who had a testimony fall short so often? How could someone who knew as Peter knew slip as often as Peter did, even to the point of an outright denial? The answer to such questions seems to lie in a conversation between Jesus and Peter at the Last Supper. Jesus said, “Simon, Simon, behold, Satan hath desired to have you, that he may sift you as wheat: But I have prayed for thee, that thy faith fail not: and when thou art converted, strengthen thy brethren” (Luke 22:31-32).
Because the full power and gifts of the Holy Ghost were not given until the day of Pentecost, though Peter had a testimony, he was not yet fully converted. After the resurrection of the Lord and after Pentecost and the accompanying baptism by fire, Peter and the Twelve walked in a new light. We need only to read the opening chapters of the Acts of the Apostles to witness a transformation in the man Peter. He is bold and certain and solid in his ministry—the permanent and indelible impression was planted, and the Holy Ghost had made Peter into a new creature, a new creature alive in Christ.
As Peter and John walked through the Gate Beautiful on the way to the temple (Acts 3), they passed a lame man who begged alms daily. President Harold B. Lee described the touching scene: “Here was one who had never walked, impotent from his birth, begging alms of all who approached the gate. And as Peter and John approached, he held out his hand expectantly, asking for alms. Peter, speaking for this pair of missionaries—church authorities—said, ‘Look on us.’ And, of course, that heightened his expectation. ‘Then Peter said, Silver and gold have I none; but such as I have give I thee: In the name of Jesus Christ of Nazareth rise up and walk.'”
President Lee continued: “Will you see that picture now of that noble soul, that chiefest of the apostles, perhaps with his arms around the shoulders of this man, and saying, ‘Now, my good man, have courage, I will take a few steps with you. Let’s walk together, and I assure you that you can walk, because you have received a blessing by the power and authority that God has given us as men, his servants.’ Then the man leaped with joy.” Through the cultivation of the gift of the Holy Ghost, Peter was born again, converted, turned wholly to Christ and to His righteousness. Peter could now strengthen his brothers and sisters. “You cannot lift another soul,” President Lee added, “until you are standing on higher ground than he is. You must be sure, if you would rescue the man, that you yourself are setting the example of what you would have him be. You cannot light a fire in another soul unless it is burning in your own soul” (in Conference Report, Apr. 1973, 178).
Truly, a person is converted “when he sees with his eyes what he ought to see; when he hears with his ears what he ought to hear; and when he understands with his heart what he ought to understand accompanied by the gift of the Holy Ghost. And what he ought to see, hear, and understand is truth—eternal truth—and then practice it. That is conversion. . . . 
“Thou art the Christ, the Son of the Living God”
What does the average person mean when he testifies that Jesus is the Christ? Of course, it is the witness of the Spirit that counts. But what do the words Jesus and Christ mean? A brief excursion into the meaning of these two words may be useful.
Elder G. Homer Durham of the Seventy gives valuable information as to the meaning of the words Jesus and Christ.
The Oxford English Dictionary on Historical Principles (Oxford: The Clarendon Press, 1933), tells us that the word Jesus came into the English language from Middle English, adapted from the Latin Iesus,which in turn was adapted from the Greek Iesous. This in turn was adapted from the Hebrew or Aramaic word Yeshua or Yehoshua. The earlier root was Joshua. This dictionary goes on to explain that the word Joshua derived from the Jah of Jahveh, meaning that “Jehovah is salvation.” Thus, the word “Jesus” has parallel meaning withSavior. Dr. David Flusser of the Hebrew University of Jerusalem puts it simply: “Jesus is the common Greek form of the Hebrew name Joshua.” (Encyclopedia Judaica, Jerusalem: Keter Publishing House, Ltd., 1971, 10:10.)
Webster’s New 20th Century Dictionary of the English Language(Cleveland: World Publishing Co., 1968), unabridged, sets forth a comparable definition, noting that the Latin-Greek derivation from the Hebrew Joshua means literally, “help of Jehovah.” But in addition, this source states that the word derives from the Hebrew word for Lord God, he who is available to help, to save. In this sense, then, the word Jesus means simply, “God is help.” How “Save the Humans,” as the bumper sticker read? The dictionaries and the gospel give the answer.
What of the word Christ? It also comes to the English-speaking world from Middle English, derived from the Latin Christus, in turn from the Greek Christos, which meant “the Anointed”—a noun made from the past participle of the Greek verb “to anoint”.
Webster also states that the word Christ was originally Jesus’ title. Thus, proper usage of the two words in English would be as Elder James E. Talmage titled his book, Jesus the Christ. Usage and revelation have joined the two as part of a sacred, revered name.
Elder Talmage defined the two words as follows:
“Jesus is the individual name of the Savior, and as thus spelled is of Greek derivation. … In the original the name was well understood as meaning ‘Help of Jehovah,’ or, ‘Savior.’ ”
Elder Talmage emphasized that the word Christ is a sacred title, not “an ordinary … common name; it is of Greek derivation, and … is identical with its Hebrew equivalent Messiah … , signifying the Anointed One.” (Jesus the Christ, Salt Lake City: Deseret Book Co., 1973, pp. 35-36.) 
“Upon this rock I will build my church . . . and I will give unto thee the keys of the kingdom of heaven . . .”
Matthew’s is the only Gospel in the New Testament to use the termchurch (Greek ekklesi) in referring to the organized community of believers. The key reference occurs in the account of Jesus’ discussion with his disciples at Caesarea Philippi. When Peter testifies of Christ’s divine Sonship, the Master replies that this knowledge is of divine origin and then goes on to speak of the Church and its leadership.
The keys or right of presidency by which the Church of Jesus Christ was to be established, the means by which ordinances were to be performed, and the authorization to make converts and seal them into a family order were all received by the chief Apostles from Jesus and heavenly ministrants. It appears that on the Mount of Transfiguration Moses and Elijah restored the keys of the gathering of Israel, as well as the sacred sealing power (Matt. 17:1-8; compare D&C 110). There seemed to be no doubt Jesus had come to earth to do more than articulate ethical principles or even to reveal strong doctrine; the Master came as a legal administrator and reestablished the kingdom of God–the Church of Jesus Christ 
 D. Kelly Ogden and Jeffery R. Chadwick, The Holy Land: A Geographical, Historical and Archaeological Guide to the Land Of The Bible, 1990 Notes, HaMakor, Jerusalem, Israel
 Robert L. Millet, Alive in Christ: The Miracle of Spiritual Rebirth[Salt Lake City: Deseret Book Co., 1997], 96.
 G. Homer Durham, “Jesus the Christ: the Words and Their Meaning,” Ensign, May 1984, p. 14
 Joseph Smith and the Gospel of Matthew by Robert L. Millet, BYU Studies, vol. 25 1985, Number 3-Summer 1985 p. 70)