Another important passage today. We've see a couple of "I am" passages from Jesus so far, and we'll see more as we go on. Something sets this one apart though:
"The climax of the entire chapter arrives at 8:58: “ ‘I tell you the truth,’ Jesus answered, ‘before Abraham was born, I am!’ ” The seriousness of this statement is confirmed by Jesus’ preface (lit.): “Truly, truly [Gk. amen, amen] I say to you,” a phrase Jesus uses some twenty times in the Gospel. This is an absolute claim to preexistence anchored in the absolute “I am” (Gk. ego eimi) language we have already seen in this Gospel (cf. 4:26). “I am” possesses no predicate (as in “I am the bread of life,” 6:35) and so stands alone, no doubt echoing the Greek translation of God’s divine name given in Exodus 3:14. To exist before the birth of Abraham—and yet to stand here today—is the boldest claim Jesus has yet made. It recalls the affirmation of the prologue that the Word existed even at the beginning of time. His existence has been continuous since his life is completely drawn from God’s eternal life."  (emphasis mine)
This is Jesus undeniably claiming to be God by taking the divine name "I AM". How do we know this? The response of the crowd. As soon as Jesus makes this statement, they pick up stones to stone Him to death.
So why is this important? There are many people who like Jesus - they think He was a good teacher or whatnot. There are many who say that He never claimed to be God. This is evidence to the contrary. If you ever need a verse to show that Jesus claimed to be God, this is the one. And so we are left with few options. As always, Clive Staples Lewis says it the best:
"I am trying here to prevent anyone saying the really foolish thing that people often say about Him: I’m ready to accept Jesus as a great moral teacher, but I don’t accept his claim to be God. That is the one thing we must not say. A man who was merely a man and said the sort of things Jesus said would not be a great moral teacher. He would either be a lunatic — on the level with the man who says he is a poached egg — or else he would be the Devil of Hell. You must make your choice. Either this man was, and is, the Son of God, or else a madman or something worse. You can shut him up for a fool, you can spit at him and kill him as a demon or you can fall at his feet and call him Lord and God, but let us not come with any patronising nonsense about his being a great human teacher. He has not left that open to us. He did not intend to." 
 Lewis, C.S., Mere Christianity, London: Collins, 1952, 54-56.