Q:

Why does Paul call Peter "Cephas" in his writings?

In John 1:42 Jesus called Peter as Cephas. 

Jesus looked at him and said, “You are Simon son of John. You will be called Cephas” (which, when translated, is Peter).

But throughout the Gospels, Peter was called as Peter and his books also called in his name I Peter and II Peter.
But why did Paul call Peter, Cephas (Galatians 2:7-14, I Cor. 1:11-13, I Cor. 3:21, I Cor. 9:5 and I Cor. 15:5)? In some occasions he also calls him as Peter. When I asked my mom, she said Paul was rebuking Peter in the name Cephas. Is this true? Or is calling Peter as Cephas normal?
At times, Paul and Peter had disagreements as in Galatians 2:11-16: 

11 When Cephas came to Antioch, I opposed him to his face, because he stood condemned. 12 For before certain men came from James, he used to eat with the Gentiles. But when they arrived, he began to draw back and separate himself from the Gentiles because he was afraid of those who belonged to the circumcision group. 13 The other Jews joined him in his hypocrisy, so that by their hypocrisy even Barnabas was led astray.
   14 When I saw that they were not acting in line with the truth of the gospel, I said to Cephas in front of them all, “You are a Jew, yet you live like a Gentile and not like a Jew. How is it, then, that you force Gentiles to follow Jewish customs?
15 “We who are Jews by birth and not sinful Gentiles 16 know that a person is not justified by the works of the law, but by faith in Jesus Christ. So we, too, have put our faith in Christ Jesus that we may be justified by faith in Christ and not by the works of the law, because by the works of the law no one will be justified.

How was their relationship as fellow apostles? 

A:

Peter means "stone" in Greek, while Cephas is "stone" in Aramaic.
That verse is confusing in the NIV; the KJV is clearer:

John 1:42 (KJV)
42And he brought him to Jesus. And when Jesus beheld him, he said, Thou art Simon the son of Jona: thou shalt be called Cephas, which is by interpretation, A stone.

Christ was speaking in Aramaic, and would have called Simon "Cephas." John wrote his gospel in Greek, so he included the note for his readers that "Cephas" meant "a stone."

A:

This is a very easy question to answer, and I'm not sure why the obvious hasn't been clearly enough stated: Πέτρος (Petros) is simply Greek (naturally, because the New Testament was written in Greek) for the name Jesus actually gave him in Aramaic: כיפא (Kefa—Rock).
In Greek, Kepha is transliterated (not translated) as Κηφᾶς (Kephas—a representation of the Aramaic name with the necessary Greek grammar 'salting'), but is sometimes translated into the Greek, and so you see Πέτρος (Petros—the equivalent of כיפא in Greek; Rock, Stone).

St. Paul might have used kepha to emphasize St. Peter's role (as his name signifies) as the Rock or leader of the Church (such as emphasizing that he had to even rebuke the 'Rock' of the Church, when it was called for).
We see that Jesus, when He first called Peter (not later in His ministry is he given the new name, as is a common misconception) to be His disciple, gave him a new name. As we know from Scripture, this always signifies a new role or significance for the person: e.g. Abram (high father) to Abraham (father of many).

John 1:41
He findeth first his brother Simon, and saith to him: We have found the Messiah, which is, being interpreted, the Christ. And he brought him to Jesus. And Jesus looking upon him, said: Thou art Simon the son of Jona: thou shalt be called Cephas [still a Greek transliteration of Aramaic Kepha], which is interpreted Peter [i.e. which in the Greek in which this is written, is Petros].

Then later, Jesus reveals the reason behind the name change:

Matthew 16:16-19
Simon Peter answered and said: Thou art Christ, the Son of the living God. And Jesus answering, said to him: Blessed art thou, Simon Bar-Jona [Ar. for 'Simon, son of John']: because flesh and blood hath not revealed it to thee, but my Father who is in heaven. And I say to thee: That thou art Peter [Petros*]; and upon this rock [petra*] I will build my church, and the gates of hell shall not prevail against it. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven.

* A name for a man cannot be the bare (feminine) word for rock in Greek (Petra) and so is made into a masculine name form: Petros. It has nothing to do with a big rock and a small rock; such is nonsense and destroys the grammatical connection of 'this' with 'rock'—'Thou art Peter. And in other news, something nothing to do with little rocks, upon this huge cliff I will build by Church.'
He is making Peter the 'rock' upon which His Church is built in the sense that He is entrusting him with the authority of 'steward' in His Kingdom of Heaven on earth. In saying what He does the way He does He harkens back to a passage in Isaiah where the same language is used of Eliakim:

Isaiah 22:20:-23
And it shall come to pass in that day, that I will call my servant Eliakim the son of Helcias, and I will clothe him with thy robe, and will strengthen him with thy girdle, and will give thy power into his hand: and he shall be as a father to the inhabitants of Jerusalem, and to the house of Juda. And I will lay the key of the house of David upon his shoulder: and he shall open, and none shall shut: and he shall shut, and none shall open. And I will fasten him as a peg in a sure place, and he shall be for a throne of glory to the house of his father.

This investment with authority (seen throughout Acts etc) is the sense in which Peter, the exemplar in the profession of the Faith in Christ as Messiah and Lord, the Son of the Highest, is the 'rock' of the Church. The hinge around which the unity of Faith is justly pivoted, per its institution this why by the Lord Jesus Himself.

St. Paul cleared up any disputes he had with St. Peter as we see in Acts 15.

