The four entries in this series are now available as one paper on the Studies page. I encourage you to head over there to get the full picture.
When we come to the term, "evangelist," we are dealing with a term used far less often than "apostle" or "prophet." Euangelistes occurs only three times in the New Testament: Acts 21:8 refers to "Philip the Evangelist, one of the Seven.". In 2 Timothy 4:5, Paul encourages Timothy to "do the work of an evangelist." The third reference is in the verse presently under discussion in this series, Ephesians 4:11.
Etymologically, euangelistes means "one who preaches the Gospel." Most often, it is Jesus and the apostles who preach the gospel (euangelizo); presumably, an evangelist would be someone who preached the gospel and who didn't fit into one of the other recognized ministries. If we look at the example of the one person actually named an "evangelist" in the New Testament, we can gain a better perspective of what this office entails.
As mentioned above, Philip is first mentioned in connection with the Seven who had been chosen to assist the Apostles in Acts 6:3. We next meet him in the aftermath of the persecution in Jerusalem that began with the stoning of Stephen (8:1-3). "Those who had been scattered preached [euangelizo] the word wherever they went. Philip went down to a city in Samaria and proclaimed the Christ there" (8:4-5). This is the first mission to non-Jewish people recorded in Acts. Philip's ministry was extremely effective--accompanied by miraculous signs and the evident conversion (signified by baptism) of many who had previously followed a sorcerer named Simon (8:6-12). It was only after the success of Philip's ministry ("the apostles in Jerusalem heard that Samaria had accepted the word of God," 8:14) that Peter and John were sent to lay hands on the people for them to receive the Holy Spirit.
Philip was next used in the conversion of the Ethiopian eunuch (8:26-40), the first recorded conversion of a fully ethnic Gentile. God had simply directed him to go by a certain route toward Gaza, which he never reached. He met the eunuch on the way, who was already reading one of the "suffering servant" passages in Isaiah, he "began with that very passage of Scripture and told him the good news [euangelizo] about Jesus" (8:35). The eunuch asked to be baptized, Philip did so, and was immediately transported to Azotus "and traveled about, preaching the gospel [euangelizo] in all the towns until he reached Caesarea" (8:40).
So what is Philip's ministry? He goes to unreached people, preaches the gospel with effectiveness and supernatural power, baptizes people--and then moves on. The key term here is "unreached people." It appears evident that the evangelist, biblically, is yet another type of missionary: one who is called specifically to reach the unreached and whose work essentially ends with conversion. An evangelist is supernaturally empowered to bring the Gospel to the lost with the result that they come to faith in Christ. The difference between an evangelist and an apostle is that while the latter is a church-planting missionary who not only brings people to salvation but also births, nurtures, and provides subsequent oversight to communities of faith, the evangelist's work is more specifically to introduce the gospel to people and to bring them to a saving knowledge of Christ. It may be the case, as it evidently was in Samaria, that the evangelist spearheads the work in an unreached area and the apostle comes in subsequently to establish and ground the work.
Or one person may fulfill both roles, as the Apostle Paul evidently did, and as did the other New Testament character associated with the term, "evangelist," Timothy. Included among the apostles in 1 Thess. 2:6, Timothy was appointed by Paul to stay behind in Ephesus while Paul traveled to Macedonia (1 Tim. 1:3), and in 2 Timothy, written when Paul was expecting to be martyred for his faith, Paul exhorts Timothy, in the midst of doctrinal confusion and rejection of the truth, to "keep your head in all situations, endure hardship, do the work of an evangelist, discharge all the duties of your ministry" (4:5). Timothy is never exactly called an evangelist, as is Philip; he is exhorted to "do the work of an evangelist"--presumably, one of Timothy's gifts is to preach the gospel to unbelievers and bring them to faith. Rather than being wholly distinct offices, we can see that the types of ministry that God had given to the church may be somewhat fluid; C. Peter Wagner profitably discusses a "gift mix" rather than each person having only one specific gift. But Philip most clearly embodies the evangelist qua evangelist: a missionary who reaches the unreached and brings them to faith in Christ.
What appears to be clear is that the evangelist, biblically, is not what is usually termed an evangelist today--an itinerent speaker who goes from church to church, possibly with a message of salvation, but largely to excite, motivate, or possibly teach or otherwise minister to believers. The modern-day evangelist might better be termed a "revivalist." This is not to say that that gift is invalid; it is merely to say that when Paul writes that "it was he who gave some... to be evangelists," (Eph. 4:11), he had in mind more of a missions emphasis and less of a revival emphasis than we normally associate with the term. In some sense, "evangelist" is to "apostle" what "preacher" is to "pastor": the former term boils the much larger and complex role of the latter term down to the essence of proclamation. Just as with "apostle," in "evangelist," we are dealing essentially with a transliteration of a Greek word. A native Koine Greek speaker would have heard "good news" in the very term, "evangelist," and that good news is very specifically the message of salvation through the death and resurrection of the Lord Jesus.