Church Foundation

The Catholic Church teaches that Peter was the first pope and appointed as such by Jesus, that Peter was the rock on which the church of God on earth would be built.

The Greek Words Involved
This is however, a very clear and wrong interpretation of Matthew 16:18. Just in looking at the English translation, it's not as clear, but when looking at the original Greek, it becomes apparent what is happening:

Matthew 16:18-19 And [de] I say [lego] also [kago] unto thee [soi], That [hoti] thou [su] art [ei] Peter [Petros], and [kai]upon [epi] this [taute] rock [petra] I will build [oikodomeo] my [mou] church [ekklesia]; and [kai] the gates [pule] of hell [hades] shall [katischuo] not [ou] prevail against [katischuo] it [autos]. And [kai] I will give [didomi] unto thee [soi] the keys [kleis] of the kingdom [basileia] of heaven [ouranos]: and [kai] whatsoever [hos] [ean] thou shalt bind [deo] on [epi] earth [ge] shall be [esomai] bound [deo] in [en] heaven [ouranos]: and [kai] whatsoever [hos] [ean] thou shalt loose [luo] on [epi] earth [ge] shall be [esomai] loosed [luo] in [en] heaven [ouranos].

Catholics want to say Peter is "the rock" being mentioned in Matthew 16:18. However, they are actually two different words. Peter is from the Greek word "Petros" while the word "rock" is "petra". Peter you may recall was originally called Simon, but Jesus renamed him "Cephas", which means "stone":

John 1:42 And [kai] he brought [ago] him [autos] to [pros] Jesus [iesous]. And [de] when Jesus [iesous] beheld [emblepo] him [autos], he said [epo], Thou [su] art [ei] Simon [simon] the son [huios] of Jona [ionas]: thou [su] shalt be called [kaleo] Cephas [kephas], which [hos] is by interpretation [hermeneuo], A stone [petros]. The word "petra" translated "rock" is according to Strong's Greek, defined as "feminine of the same as 4074; a (mass of) rock (literally or figuratively):--rock." The word "Petros" translated "Peter" is defined, "apparently a primary word; a (piece of) rock (larger than 3037); as a name, Petrus, an apostle:--Peter, rock. Compare 2786."

Peter Not the Rock
Essentially it was a play on words. Jesus was the Rock. Peter, because of some similarity, was called "little rock" or "stone".

Jesus Was the Rock
However, there is undisputable proof that Peter was not the rock being discussed. You see, that word "petra" translated "rock" in Matthew 16:18 is never once used of Peter in the entire New Testament. It is only used 14 times in the Bible, including Matthew 16:18. By looking at the other 13 times, it is apparent that it is used typically to refer, not to Peter, but to someone else:

1 Corinthians 10:4 And [kai] did [pino] all [pas] drink [pino] the same [autos] spiritual [pneumatikos] drink [poma]: for [gar] they drank [pino] of [ek] that spiritual [pneumatikos] Rock [petra] that followed them [akoloutheo]: and [de] that Rock [petra] was [en] Christ [Christos].

1 Peter 2:5-8 Ye [autos] also [kai], as [hos] lively [zao] stones [lithos], are built up [oikodomeo] a spiritual [pneumatikos] house [oikos], an holy [hagios] priesthood [hierateuma], to offer up [anaphero] spiritual [pneumatikos] sacrifices [thusia], acceptable [euprosdektos] to God [theos] by [dia] Jesus [Iesous] Christ [Christos]. Wherefore [dio] also [kai] it is contained [periecho] in [en] the scripture [graphe], Behold [idou], I lay [tithemi] in [en] Sion [Sion] a chief corner [akrogoniaios] stone [lithos], elect [eklektos], precious [entimos]: and [kai] he that believeth [pisteuo] on [epi] him [autos] shall [kataischuno] not be [ou me] confounded [kataischuno]. Unto you [humin] therefore [oun] which [ho] believe [pisteuo] he is precious [time]: but [de] unto them which be disobedient [apeitheo], the stone [lithos] which [hos] the builders [oikodomeo] disallowed [apodokimazo], the same [houtos] is made [ginomai] [eis] the head [kephale] of the corner [gonia], And [kai] a stone [lithos] of stumbling [proskomma], and [kai] a rock [petra] of offence [skandalon], even to them which [hos] stumble [proskopto] at the word [logos], being disobedient [apeitheo]: whereunto [eis] [hos] also [kai] they were appointed [tithemi].

Romans 9:33 As [kathos] it is written [grapho], Behold [idou], I lay [tithemi] in [en] Sion [Sion] a stumblingstone [lithos] [proskomma] and [kai] rock [petra] of offence [skandalon]: and [kai] whosoever [pas] believeth [pisteuo] on [epi] him [autos] shall [kataischuno] not [ou] be ashamed [kataischuno].

Simply reading the verses, it is said as plain as day, and never plainer than Paul puts it in 1 Corinthians 10:4 when he says, "and that Rock was Christ". That same word translated "rock", "petra", in Matthew 16:18 is the exact same one Paul uses here in saying it refers to Christ. Peter in 1 Peter 2:5-8 says the same thing, pointing out that we are all simply stones (small rocks) built into a spiritual house and holy priesthood to offer up sacrifices to God through Jesus Christ. Peter says the same thing, that it is Jesus, not himself, who is the "rock of offence" and "chief corner stone" that the builders rejected, who was prophesied of as such in the Old Testament:

Isaiah 8:13-14 Sanctify the LORD of hosts himself; and let him be your fear, and let him be your dread. And he shall be for a sanctuary; but for a stone of stumbling and for a rock of offence to both the houses of Israel, for a gin and for a snare to the inhabitants of Jerusalem.

Isaiah 28:16 Therefore thus saith the Lord GOD, Behold, I lay in Zion for a foundation a stone, a tried stone, a precious corner stone, a sure foundation: he that believeth shall not make haste.

Psalms 118:22 The stone which the builders refused is become the head stone of the corner.

It is, in other words, Jesus who is the Rock on which the Church was built. When Jesus spoke in Matthew 16:18 to Peter, He was essentially saying to him, You are Peter (a little rock) and on this big Rock (Jesus Himself) I will build my church, and the gates of hell will not prevail against it. It is as plain as day from reading the Bible.

Keys of the Kingdom of Heaven Given to All Disciples
As for the keys to the kingdom of heaven, and the ability to bind and loose, it was given to all disciples, not Peter only:

Matthew 18:18-20 Verily I say unto you, Whatsoever ye shall bind on earth shall be bound in heaven: and whatsoever ye shall loose on earth shall be loosed in heaven. Again I say unto you, That if two of you shall agree on earth as touching any thing that they shall ask, it shall be done for them of my Father which is in heaven. For where two or three are gathered together in my name, there am I in the midst of them.

John 20:20-23 And when he had so said, he shewed unto them his hands and his side. Then were the disciples glad, when they saw the Lord. Then said Jesus to them again, Peace be unto you: as my Father hath sent me, even so send I you. And when he had said this, he breathed on them, and saith unto them, Receive ye the Holy Ghost: Whose soever sins ye remit, they are remitted unto them; and whose soever sins ye retain, they are retained.

It is plain all the disciples were given such power, not Peter only, as seen from the above verses. Peter was not being given some special privilege, but Jesus was revealing the nature of the Church to be built on Himself, and the power He would give all the disciples.