1. When Jesus was ministering on earth, he was obviously the leader of the group. After the twelve (now eleven) disciples witnessed his resurrection, Jesus sent them as the apostles, as witnesses of the resurrection sent to spread the good news. Judas was replaced by Matthias, restoring the number of the Apostles to twelve.
Note that, according to the requirements for Judas' replacement, the apostle Paul would not be qualified as one of the Twelve. According to Acts, his replacement needed to be someone who had been with Jesus from the time of John the Baptist (Acts 1:22). So it goes without question that no one alive today could be one of the twelve apostles, since not even Paul qualified by Acts' standard.
2. There is, however, a second level understanding of an apostle we find, one that includes Paul. He understands an apostle to be someone to whom the risen Christ has appeared (1 Cor. 9:1) and who has then been sent to proclaim the good news. 1 Corinthians 15:5-8 gives us a sent of those to whom Jesus appeared: first Peter, then the twelve (apostles), then James and the other apostles. Then he was the last.
So again, there are no apostles of this sort today, or at least Paul did not anticipate there being any. The New Testament apostles were individuals to whom the risen Christ had visibly appeared and whom Jesus himself had commissioned, perhaps audibly.
We know the names of some in this company. Paul seems to include Barnabas in this list (1 Cor. 9:6; cf. Acts 14:14). In Romans 16:7 may very well include a husband-wife team among the apostles, a couple named Andronicus and Junia. If so, we should expect that they were two of those to whom Jesus appeared and that they were part of the early church's mission.
3. Is there a third layer of apostle in the New Testament? There are other people who are "sent" by someone or another. Paul calls Epaphroditus a "messenger" of the church of the Philippian church (Phil. 2:25). Here he uses the same word but not in the strong sense of the word elsewhere. Here it is used in its more ordinary sense of someone who is sent from someone to bring a message to someone else.
So in the face of the consistent New Testament use of the word apostolos, it is overwhelmingly likely that 1 Corinthians 12:28 and Ephesians 4:11 are speaking only in relation to the apostles of the early church when they speak of apostles. The foundation of the (early) church were these apostles and another group known as the prophets (Eph. 2:20). We should remember that, at least in his earliest letters, Paul did not foresee that the Lord would tarry this long (cf. 1 Cor. 7:29). He did not see himself as setting up a structure that related to the next two thousand years.
4. So what of today? There are groups that call leaders apostles today. These leaders are characterized by extreme charisma and heightened power in their ministries. God knows their hearts. Words change their meanings over time. If some Christian groups are inspired for mission by a particular use of words, may the Lord be with them. But know that these are not apostles in the biblical sense of the word.
It is also common to speak of "apostolic ministry," that is, to speak of individuals playing roles in the church today similar to the roles the apostles played in the early church. Alan Hirsch and Tim Catchim understand the function of apostles as being that of mission and church planting.  For them, those who play the apostolic role are tasked with the extension of the church and thus its overall vigor.
Interestingly, church history has more often seen the highest level of institutional leadership as the heirs of the apostles' role in the early church. That is to say, they implicitly see the role of the apostles in the earliest church as that of final authority, not least in the writing of Scripture. So for most of the last two thousand years, apostolic authority is seen to continue in Scripture and in institutionalized leadership.
Perhaps there is truth in both of these models. Yes, one of the key purposes of Scripture was for the original apostles to leave a "deposit" of right teaching (orthodoxy) after their deaths (cf. 1 Tim. 6:20). Yes, the apostles appointed leaders to serve as an authority structure for the church (cf. 1 Clement 42:4-5; 44:1-3). Yes, Christian tradition suggests that the apostles spread throughout the world spreading the good news of Jesus' resurrection and lordship. 
So if a Christian group today wants to use the word apostolic to emphasize one of these aspects, it will be in continuity with at least one aspect of the apostles.
5. Ephesians 2:20 and 1 Corinthians 12:28 highlight another important role in the early church, namely, that of prophet. There was apparently an identifiable group within the early church known collectively as "the prophets." These were individuals to whom the Lord gave revelation for the church, not least spiritual interpretations of the Old Testament about Christ.
In Acts, we see one of these prophets in action, Agabas. In Acts 11:28, he predicts that there will be a famine in Palestine. In Acts 21:11, he predicts that Paul will be arrested in Jerusalem. 1 Corinthians 14 suggests that prophecy was widely practiced in the early church, even on the level of local churches. Paul has a very favorable view of it, suggesting it is more beneficial to the church than speaking in an unknown tongue.
A thorough study of the way the New Testament interprets the Old Testament shows that the early church was largely "charismatic" in its interpretation. The Spirit revealed truths to the church through Scripture in ways that were not limited by the original meaning or context of those verses. We can imagine that many of the "spiritual" interpretations we find in the New Testament came from these prophets. Indeed, some have suggested that some of the words of Jesus in the Gospels may have come after his ascension as prophetic words from the risen Lord that came through prophets (e.g., Matt. 10:38).
Unlike the original apostles, there is nothing about the nature of a prophet in the New Testament that suggests this function had any reason to stop with the deaths of the first apostles.  In the church today, prophetic voices are those that receive words from the Lord and pass them on to the church. Certainly some pastors fit into this category.
But the very nature of the prophetic is that it often stands outside the normal, institutional structures of the church. Prophets often come unexpectedly (and often annoyingly) from the sidelines. Anyone who has a word from the Lord to speak to the church--whether it is corrective or predictive or encouraging--is functioning like a prophet today.
6. We find another role of early church leadership in Acts 15. When the question whether Gentiles can be saved comes up, it is discussed by the "apostles and elders" (15:2). So there was a group of leaders in the Jerusalem church known as the elders. No doubt following the pattern of Jewish synagogue leadership, the earliest church seems to have had a group of older individuals who provided wisdom and leadership to local Christian communities. These were probably mostly men, in keeping with the culture.
So it seems that Paul's local churches included groups of elders who provided leadership to those house churches (Acts 14:23). We should not think that this was the only leadership structure in play. It seems that there were sometimes key individuals who provided spiritual leadership. Timothy, for example, seems to have played a stronger role in the Ephesian church than just another elder. The same would apply to Titus at Crete. Names like Epaphroditus, Epaphras, and Archippus may indicate individuals with special roles in key churches.
The word "overseer" (episkopos) seems roughly equivalent to an elder. Titus 1:5-7 treats the two as the same role in the church, as 1 Peter 5:1-1 also seems to imply.  So when 1 Timothy 3 gives guidelines for overseers, it is likely giving guidelines for the elders of house churches or city wide collections of house churches.
By the early second century, the word overseer had taken on larger proportions. Ignatius seems to be the sole "bishop" (as we now might translate the word) for the whole city of Antioch in Syria. We have evidence from inscriptions that there were some women who were bishops in the first few centuries of the church.
7. Another role in the early church is that of deacon. The Greek word diakonos seems to refer to a role "serving" the local church. Although Acts 6 does not actually use this word, we can imagine that the kind of role portrayed there for Stephen and Philip is similar to the kind of role deacons served. These would be more mundane roles such as those done in churches today by trustees, volunteers, and, in fact, roles actually called "deacons" today.
Paul mentions a woman named Phoebe who was deacon of the port house church at Cenchrea, four miles southeast of Corinth. In Philippians 1:1, Paul indicates that the church of Philippi also had deacons. 
8. A word of caution is in order here. There are some in the church who, sometimes condescendingly, criticize churches that do not use these names for various leadership roles in the church today. Two important qualifications should be mentioned. The first is that there may have been varying leadership structures in the early church. We should not assume that all churches were led in the same way.
We do not find a single language in the New Testament. For example, the word deacon does not appear in Acts. Paul never uses the word elder in his primary letters. Some churches were probably more "charismatic" than others, others more structured, just like people have different personalities.
Perhaps more to the point, the structures of the early church were "incarnated" structures. That is to say, they fit the cultural dynamics of the day. It is foolish to think that the same leadership structures will be equally effective in all places, cultures, and times. Doing things the way they did them then will often yield quite different outcomes in a different time and place.
The church thus needs to do its best to incarnate the right values into forms that fit different wineskins. The early church had leaders. It had an authority structure. It developed institutions in some places (like the authority of apostles and elders in Jerusalem). It also had room for the Holy Spirit and more charismatic leadership that came "off the institutional grid," so to speak.
These are two important poles for the church today: institution and charisma, apostle and prophet. Without the prophet, institutions become stale and bureaucratized. Without the apostle and council of elders, there is no clear official answer in controversy.
Meanwhile, there must be local leaders and there must be "get it done" type people. Although it is not a neat division, the broader church apostles and prophets are like the strategic leaders of the church.  The local elders and overseers are like the tactical leaders of the church. Then the deacons are like the operational leaders of the church.
Of course good strategy can come from anyone, as can good tactical and operational insights. In the modern church, if you would, we can line up these types of leaderships roughly with those who lead, those who manage, and those who administrate.
Next Week: Pastor as Leader 17: Historical Church Structures
 E.g., Alan Hirsch and Tim Catchim, Permanent Revolution: Apostolic Imagination and Practice for the 21st Century Church (San Francisco: Jossey-Bass, 2012), 8.
 Except for what we know of Paul and hints of Peter, the New Testament really gives us almost no information on what happened to the other apostles. Our sense that the apostles spread out all over the world comes from tradition rather than the Bible.
 It is horribly bad exegesis to interpret 1 Corinthians 13:8 this way. Paul is here either talking about specific prophecies (rather than prophecy in general) or if he had prophecy in general in view, he would be referring to the eschaton and the return of Christ, which has of course not yet happened. It is true that 2 Peter 3:2 may speak of these early church prophets in the past tense, but 2 Peter has in mind the earliest predictions of Christ's immanent return.
 The word variously translated as "oversight" or "watching" is the verb form of the word for overseer.
 It has long been tantalizing to me that Timothy is referred to as a diakonos in 1 Timothy 4:6, although most translations are probably right to translate it as "minister" or "servant" in this instance.
 See here for this distinction I took from Bob Whitesel.
Leadership in General
- The Mission, Great and Small
- Identifying Mission
- Casting Vision
- Vision Statements
- Thinking Missionally
- Evaluating Strengths
- Identifying Core Values
- Setting Goals
- Leading Change
- Evaluating Progress/Resetting Goals