Some of the favorite passages used by Calvinists to argue for unconditional election come from John 6 and 10. My purpose here is to present a very quick explanation of the correct interpretation of these verses.
Specifically in John 6:37, 39, Jesus says, "37 All that the Father gives Me will come to Me, and the one who comes to Me I will by no means cast out...39 This is the will of the Father who sent Me, that of all He has given Me I should lose nothing, but should raise it up at the last day."
It's made clear in the context who the "all that" Jesus is referring to is in v. 40 and 47. Believers as a whole! Furthermore it is shown that those who have truly listened to the teaching of the Father will come to Christ in v. 45.
Verse 65, which says, "Therefore I have said to you that no one can come to Me unless it has been granted to him by My Father," is easily understood to show that unless someone has been enabled by the Spirit of God, they cannot come to Christ. But unconditional election does not follow from that, and is shown to be false in light of the context of not only this chapter, but the whole of the Johannine witness.
This takes us to ch. 10 and the sheep that follow Christ. The sheep that follow are the faithful Jews who truly belonged to the Father and were, in light of that, being transferred to Jesus. When Jesus came, all of the truly God-fearing Jews naturally recognized Jesus as one with the Father and went to Him. This makes it clear why the Jews who were not following Jesus were not believing; "26 you do not believe, because you are not of My sheep, as I said to you." They were not believing in Jesus because they did not truly believe in the Father to begin with.
"27 My sheep hear My voice, and I know them, and they follow Me." Those who did follow Jesus did so naturally, because they knew Jesus was from the Father. This also makes sense with Jesus inserting the fact that He and His Father are one in v. 30. He's not just randomly stating it, He's explaining why His sheep recognize Him, because He and the Father are one!
For an extremely detailed exegesis, check out the article found here.