John 1: 35-42, Three Disciples Introduced to Jesus

Jesus has just been baptized by John. Jesus will also baptize people, but his baptism will be different.

John 1: 35-42, Andrew, another, and Peter
The next day John was there again with two of his disciples. 
36 When he saw Jesus passing by, he said, “Look, the Lamb of God!”

37 When the two disciples heard him say this, they followed Jesus. 
38 Turning around, Jesus saw them following and asked, “What do you want?”

They said, “Rabbi” (which means “Teacher”), “where are you staying?”

39 “Come,” he replied, “and you will see.”

So they went and saw where he was staying, and they spent that day with him. It was about four in the afternoon.
John immediately identifies Jesus as the "Lamb of God" to two of his disciples. The "Lamb" apparently refers to the sacrificial lamb of the sacrifice in the Holy of Holies on Yom Kippur.  John's identification of Jesus is intended as instructions to the two disciples and they respond by switching their allegiance to Jesus.

Jesus asks their intentions and they make it clear they want to follow him by calling him "Rabbi" and then asking where he is staying.  In what appears to be a short conversation, they switch their allegiance to Jesus and follow him from there own.

The writer identifies a certain time of day, as if he were there.

John 1: 40-42, Andrew, another, and Peter
Andrew, Simon Peter’s brother, was one of the two who heard what John had said and who had followed Jesus. 
41 The first thing Andrew did was to find his brother Simon and tell him, “We have found the Messiah” (that is, the Christ).
42 And he brought him to Jesus.

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

Who was the second disciple, the one with Andrew?  We are not sure but one reasonable conjecture is that it was John, the author of this letter.  (There is a stackexchange discussion of this here.) I find that logical given the details, such as time of day, provided by the author. The author also has a tendency, throughout the gospel, to leave unidentified a witness who is clearly intended to be himself.

What is the significance of the name change, from Simon to Peter? (Cephas/Peter means "rock".)  Jesus has plans for this "rock".