There is much debate as to the role of human responsibility and the sovereignty of God when discussing the topic of evangelism. Some people would argue that because we are commanded by God to take the gospel of Jesus Christ into the world, that our human responsibility somehow outweighs God's sovereignty and that in essence, God really is not completely and totally sovereign in the universe. However, on the flip side of this, some people believe with all pure intentions that God's sovereignty is total and complete and because it is so, evangelism is somehow not needed, because God already knows who He is going to save and that we cannot change the outcome, no matter what we do. So some believe that we as Christians can sit back and not bother to evangelize. In fact, this is a problem for those who reject Calvinism and the doctrine of election. If God has already chosen His elect, then Christ's command to evangelize is obsolete. Therefore, some people reject the doctrine of election based on this reasoning.
Now, I don't pretend to know the answer to this apparent contradiction, but we know that the Bible teaches both human responsibility and the doctrine of election. There is no doubt in my mind that God is totally sovereign in the salvation of His elect. And at the same time, the Bible teaches that we as Christians have a responsibility to share the gospel with those who do not know Christ. God has chosen to work through His people to bring the lost to Himself. Because of this chosen method, it is our responsibility to be about the business of evangelism, even though God is completely sovereign in His election of His people. To ignore one doctrine over another would be a mistake.
One might ask, then, how do we deal with this seeming contradiction? First of all, we have to realize that it just appears to be a contradiction, not that it really is a contradiction. Just because our finite minds cannot comprehend how both of these doctrines work together does not mean they they are at war with one another. In fact, Spurgeon was once asked if he could reconcile these two truths (human responsibility and the sovereignty of God) together. His reply was, "I wouldn't try. I never reconcile friends." In his mind, whether or not he comprehended how these two truths work together, he viewed them as friends. He saw that these two ideas were not opposed to one another, rather they were working together.
Ultimately, this mystery is a beautiful thing, because it shows us that God's ways are higher, much higher, than our ways. If we could somehow comprehend everything about God, He would not be the one, true God. He would be a god fashioned in our own image, and that would be a terrible thing indeed. What we must do is celebrate the mystery between the two truths and joyfully accept that there are things about God that we do not understand.
Finally, we take these truths with us as we take the gospel into the world through obedience to Christ's command in Matthew 28. We can be bold as we share the gospel because we know that success or failure is not based on the results that come about as the outcome of our sharing, because we are not responsible for the results. God is responsible for the saving of lost souls. We are responsible for being obedient to what God has called us to do by sharing the gospel with the lost. And even if we never see a person saved in our lifetime, we can rest assured that we are not failures. Our purpose is namely to share the glory of God to the world. God's purpose is to open the eyes of the world to His glory. This is a truth that frees us to be bold in our responsibility to evangelize the world.
Two wonderful books to read on this topic of the relationship between human responsibility and God's sovereignty in evangelism are J. I. Packer's "Evangelism & the Sovereignty of God" and John Piper's "Let the Nations Be Glad: the Supremacy of God in Missions." I would highly encourage everyone to pick up these two books.
