Evangelism is sharing the gospel (good news) message of Jesus Christ with those who do not believe, calling them to repent of their sins and become followers of Jesus. Jesus commands that all believers preach the gospel in Matthew 28:19-20, known as the Great Commission:
Therefore go and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, and teaching them to obey everything I have commanded you. And surely I am with you always, to the very end of the age.
Christians can accomplish personal evangelism to people they come in contact with everyday (family, friends, neighbors, co-workers, etc.) with not only words but actions - reflecting Christ in their lives. The church accomplishes the spread of Christianity through world-wide missions; the work of missionaries.
"A summary of what Christ aims to do through you and me when we testify to the truth of the gospel. He aims to open the eyes of the spiritually blind; he aims that they would then turn from the darkness of sin to the light of righteousness; he aims that they would turn from the power of Satan who can only hold them by lies, and come to God. He aims that all their sins be forgiven. And he aims that by faith, not legalistic burdens, they would join the saints in the pursuit of holiness. O what great, eternal things God aims to do through us as we go and tell about the truth of Jesus Christ." - John Piper