Peter the Apostle
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According to Christian tradition Peter the Apostle (c. 1-64 CE), called Simon before his conversion, was the first disciple recruited by Jesus Christ, along with his brother Andrew. The gospels claim he was present at many of the more significant events mentioned in the Bible and was a central character in Acts of the Apostles. Traditionally he was the first pope.
Like discussions about Jesus, the question of a historical Peter is met with a variety of answers. Most scholars accept that the Peter who led the church beside Paul was an actual person, and possibly the Peter who himself was a witness to the events of Jesus' life (whatever they be, historically). He is frequently mentioned by Paul in his writings, founding churches with Paul, and acting as the bishop of those churches. It is highly unlikely that Paul's letters would mention a person who did not exist, as his churches would say "Huh? There's no such man".
That said, one question often brought up is this: Is the Peter who led a very Jewish sect of Christianity, the same as the Peter who would later stand by Paul (a Hellenist) and be bishop of his Churches, and is either of these actually Simon Peter, the "right hand" of Jesus?
Leader of the apostles
In addition to his presence at key events, several specific statements made in the Bible are identified as indicating Peter's leadership role.
As further evidence of Peter's right to lead it is noted he was the only apostle to enter Jesus' empty tomb in Luke 24:12. This verse is omitted, considered interpolated, by Revised Standard Version and other translations. Peter is the first to witness the resurrection appearance in Luke and First Corinthians. It has been suggested that these verses were also interpolated. In addition, it should be noted (in context of Peter being seen as fit to lead) that when Jesus had been arrested, Peter denied three times being associated with him.
In John 21:15-17, Jesus tells Peter to "feed my sheep," a statement that has been interpreted as a request that Peter take responsibility for Jesus' followers. However, since this chapter is a later addition this statement cannot be considered historical.
The notion of the primacy of Peter is not based on evidence. In fact, it would seem to contradict other statements made by Jesus in the Bible. For example Mark 9:35 states, regarding Jesus, "And he sat down, and called the twelve, and saith unto them, If any man desire to be first, the same shall be last of all, and servant of all." (KJV)
The primacy of Peter is essential to the claim of superiority by the Church of Rome.
Traditionally Peter was the first Bishop of Rome, from about 30 or 53 CE (depending on the source) until his death, and thus the first pope. Second century church historians assert that Peter had a hand in the founding of the church in Rome, although these statements are far from proclaiming Peter the bishop. The first references to Peter as the actual bishop might be in varied works of Saint Jerome in the late fourth century.
However, consider that, despite allegedly holding the title of bishop in Rome, Peter is not mentioned once in Paul's canonical letter to the Epistle to the Romans and especially in Chapter 16, which lists many people associated with the church there. In addition there is good evidence that there was no position of bishop in the Christian Church until well into the second century.
In short there is insufficient evidence to compel the belief that Peter was the first pope. There is no evidence that Peter ever led the church in Rome or that the church in Rome led Christendom. The primacy of Rome apparently developed gradually over several centuries and does not date back to Peter. 
Per the Liber Pontificales, Peter was the first Bishop of Antioch, from 37 to 53 CE; however due to its late (ninth century) date the accuracy of this work is necessarily suspect.
Cephas is a person mentioned by Paul in his letter to the Galatians and first letter to the Corinthians. The fact that traditionally Cephas is believed to be the same person as Peter is evidenced by John 1:42. If it can be demonstrated that Peter and Cephas were the same person then that would establish Peter was a person known to Paul and thus provide evidence for his historicity.
As evidence that Peter and Cephas were different people, consider 1 Corinthians 15:5, which states, "And that he was seen of Cephas, then of the twelve." (KJV) The author of this verse believed that Cephas was not one of the twelve (apostles) and therefore not Peter. Also in Galatians Chapter 2 both Peter and Cephas are mentioned with no indication that they are the same person. In addition, the apocryphal Epistula Apostolorum contains the phrase "We, John, Thomas, Peter, Andrew, James, Philip, Bartholomew, Matthew, Nathaniel, Judas Zelotes, and Cephas," which identifies Peter and Cephas as distinct persons.
This is embarrassing. "Cephas" is just a rendering of the word "Peter" in Greek. All of the Greek manuscripts have the same word for "Cephas" and "Peter". The translators just suck.
The apocryphal Epistula Apostolorum seems to be referring to Simon "the zealot" as Cephas for some reason, probably because the other Simon changed his name to Cephas and they got confused.
Peter is credited with authorship of the First Epistle of Peter. Although challenged, this attribution is corroborated by late second century reporters.
Although the Second Epistle of Peter claims in the first verse to have been written by Peter, scholars are unanimous that it was in fact written anonymously in the second quarter of the second century with its authorship merely attributed to Peter in order to lend it credibility. Likewise the apocryphal Gospel of Peter and the Apocalypse of Peter claim Petrine authorship but were in fact written in the second century, long after Peter's death.
The Apostle Peter is said to have died in 64 CE from crucifixion. The manner of death was attested to in vague references by Christian authors writing long after the event. The apocryphal Acts of Peter, written about 100 years after Peter's death, attests to him having been crucified upside down. As a result an upside down cross became known as a Cross of St. Peter, which remains the symbol of the pontiff today, although Aleister Crowley co-opted this to be an anti-Christian symbol by arguing that someone on an upside-down cross would be farthest from the light.
John 21:18-19 is said to foreshadow Peter's crucifixion. It states, in part, "but when thou shalt be old, thou shalt stretch forth thy hands, and another shall gird thee, and carry thee whither thou wouldest not. This spake he, signifying by what death he should glorify God." Obviously this prophecy was penned with the knowledge that Peter had already passed on and, in addition to being vague, since Chapter 21 is acknowledged to have been a later addition to the Book of John, was written well after Peter's death.
Nonetheless one may conclude that there is insufficient evidence to compel the belief that Peter was crucified.