Keep in mind that Our Lord spoke Aramaic for his day to day chats with the Apostles. While Greek was well known, most Jews used the Aramaic language when speaking to fellow Jews.
Now, in Aramaic, the word for "Rock" is "Kephas" sometimes spelled "Cephas". When the Scriptures were later translated to Greek, the translator attempted to keep the word play evident in the Aramaic and thus Kephas become "Petros" when talking about the guy, and "petra" for the second instance of "rock" since it's normally a female word. The translator had to use the masculine form for Peter's name since well... Peter's a guy.
Anyways, let's look at that passage following.
You'll notice a few important things here. First, Christ tells Peter/Kephas/Cephas that it was God the Father who revealed to him who Jesus was. Second, Christ renames him! Looking in the Scriptures, whenever God gives someone a new name, it's indicative that they are to have a VERY important role to play. Abram became Abraham, Jacob becomes Israel, and now, Simon becomes Kephas.
Following this renaming, Our Lord then says that He is founding His Church on Kephas. (You are rock... on this rock...). He then also says that He is giving Kephas the keys to the kingdom of Heaven. This is a very symbolic gesture. In a sense, He's appointing Kephas to be a sort of Prime Minister, able to act in Jesus' name and with His authority. We even see it's usage today, often when someone does something extraordinary, they're given the symbolic keys to the city. While today it's merely a symbolic gesture, in those days, cities often did have a literal key and the guy who got the key was literally able to open/shut the city! So this is a VERY big deal.
In short, there's plenty of evidence that Christ did indeed found His Church on Peter(Kephas) and that he was entrusted with authority as Christ's prime minister of sorts.