The New Testament does not articulate a clear leadership structure for a local assembly of believers. That is not to say that such churches did not have a fairly common structure. We may catch glimpses of one from time to time, like when Paul greets "overseers and deacons" at Philippi (Phil. 1:1). Paul calls himself and Barnabas "apostles" (e.g., 1 Cor. 9:1-6). 1 Peter refers to "elders" (1 Pet. 5:1), and 1 Timothy has a list of qualifications for an overseer or a deacon (1 Tim. 3:1-13).
But we are left to sort out exactly what these roles were and whether they were relatively uniform throughout early Christianity. Or did the leadership structures of local assemblies and Christian synagogues actually vary somewhat from place to place. It seems impossible to answer such questions definitively, although we can make educated guesses. When all is said and done, we must ultimately recognize that even if we knew for sure how leadership took place, we would still only have descriptions, not prescriptions for how churches should structure their leadership today.
Perhaps the easiest place to begin is with the category of apostle, a person "sent" on a mission. We find the word used in more than one way in Acts and Paul's writings. On the one hand are "the Twelve." The book of Acts in particular considered the twelve apostles in a special category of apostle, one for which Paul himself would not have qualified. In Acts 1, the earliest believers replace Judas with Matthias so that the number of twelve remains intact (1:12-16).
The qualifications for this role were that a person had been with Jesus from the time of his baptism up until the time of the resurrection (1:21-22). Acts implies that more than one person might have fit into this category, but it is Matthias who is chosen. Some have speculated that the early church made a bad decision here, that Paul should have been Judas' replacement.  But Paul would not have qualified as Judas' replacement given the criteria. He apparently had no encounters with Jesus during his earthly life. 
At the same time, both Paul and Acts also use a broader definition of a "sent one," an apostle. Acts 14:14 calls Paul and Barnabas apostles in this more general sense. And when Paul is defending his rights as an apostle, his criteria seem to be 1) that one has seen the risen Lord and 2) that the risen Lord has commissioned you as a special representative of it (cf. 1 Cor. 9:1). It is in this sense that he considers himself and Barnabas to be apostles (e.g., 1 Cor. 9:5-6). And it is perhaps in this sense that he considers the husband-wife pair of Andronicus and Junia to be apostles (Rom. 16:7).
There are some Pentecostal traditions today that use the word "apostle" of some of its leaders, many of which use the word Apostolic in their name. On the one hand, it is hard to find fault with the idea that certain individuals believe themselves to be called and sent by God in some special sense. At the same time, the burden of proof is on anyone who would suggest that we have apostles today of the same sort Paul or Barnabas were. Could the risen Christ appear to someone today in the manner he appeared to Paul? Certainly.
But it is not clear that any of those who call themselves apostles today would claim to have seen the risen Christ in the same way as Paul saw him. Similarly, Paul seemed to have considered himself the last of the apostles (1 Cor. 15:8-9). With some three years separating himself from the resurrection appearances he narrates, he considers himself to have been a "miscarriage" and uses the word "lastly." By implication, he does not believe any new apostles have come on the scene in the over twenty intervening years. You can see how extremely "untimely born" would be someone claiming that suddenly, two thousand years later at the beginning of the twentieth century, a whole slough of new apostles have suddenly started to flow again!
A second category of importance in the early church seems to be that of elder. In the Methodist tradition, ministers have historically been called "ordained elders." Presbyterian and Congregationalist traditions also have a category they call "teaching" elders that understands the word in this way (cf. 1 Tim. 5:17). No doubt this use of the word relates to 1 Peter 5:1, where Peter calls himself a "fellow elder" with the elders that are leaders in the churches to which he writes.
However, the Presbyterian sense of a group of "ruling elders" in a local church probably comes closer to what the New Testament usually meant when it referred to elders. Indeed, the word "Presbyterian" is a reflection of the Greek word for elder, presbyteros. We know that members of the Jewish ruling council were considered elders (e.g., Acts 6:12; 24:1). We can imagine that synagogues throughout the Diaspora were also structured in this way. Acts refers to the leaders of the Jerusalem church beyond James and the apostles, "elders" (e.g., 15:6). So it is no surprise to hear Acts tell us that Paul and Barnabas appointed "elders" in every town (14:23).
The word elder of course refers to an older person (e.g., 1 Tim. 5:1), and so it seems overwhelmingly likely that any elders in the early church were older. We can imagine that such elders tended to be male, but we have no evidence that a woman could not be an elder in a church. The matter of Paul's churches in particular is a question. If they had elders, it is hard to imagine that a Priscilla (e.g., Rom. 16:3; Acts 18:26) or a Phoebe (Rom. 16:1) would not have been on such a body at the appropriate age.
It is, however, at least a matter of debate whether Paul's churches were structured in this way. Our personal inclination is to consider overseers (e.g., Phil. 1:1) as synonymous with elders and thus conclude that Paul's churches probably did have elders as leaders. But it is significant to notice that Paul himself does not use the word "elder" in any of his unquestioned writings. The word appears only in 1 Timothy 5, and 1 Timothy differs enough from the thinking and categories of Paul's earlier writings that most consider it pseudonymous, written to convey Paul's authority to a context several decades after his death. 
As is often pointed out, the operations of Paul's churches, at least the one at Corinth, seem to have proceeded in much more of a "charismatic" than "presbyterian" way. Indeed, the worship at Corinth was so "spirit" oriented and open in its participation that it had apparently disintegrated into chaos. Everyone had a hymn or a prophecy or a lesson or spoke in an unknown tongue/language or had an interpretation of someone's unknown language. Paul tries to steer them out of this chaos. Tongues must have interpretations. Two or at the most three with prophecy and tongues-speaking, and one at a time.
So while Paul's church probably did have appointed leaders, the Spirit element seems to have dominated, at least at Corinth. "Do not put out the fire of the Spirit," he tells the Thessalonians, "and do not despise prophecy" (1 Thess. 5:19-20). This tension between the Spirit and structure seems to have been one of the structural conflicts of the first century. We likely see it reflected in the words of Matthew 7:21-23. Here are pictured exorcists and prophets who do not make it into Jesus' kingdom. 3 John seems to reflect a conflict between a travelling evangelist (Demetrius) is rejected by a powerful local leader (Diotrephes). And 1 Timothy 3's attention to structure may very well be a response to such travelling teachers (cf. 2 Tim. 3:5-9).
Paul thus pays little attention to church structure in his commonly agreed writings. Only as his writings look to the period after his death does church leadership become an issue. This in itself is a significant observation. While Paul was alive, he was the final authority as an apostle and father to his churches. He probably did leave leadership behind, but a more pneumatic, spiritual kind of environment seems to have more been the normal mode of operation. Paul does not use established leadership as the solution to church conflict.
Acts, on the other hand, does. Acts presents a very orderly and structured church with a fairly clear chain of command. The subjugated Paul of Acts is not the free wheeling apostle of his own writings. Where we are headed with these observations is, once again, that there is no absolute church structure that the New Testament prescribes. We can read between the lines to find a description, and even this description implies some diversity of focus. The New Testament thus provides models from which we may choose. But it does not tell us how to structure our churches today in terms of specifics.
We have two more categories of early church leadership to consider, which we must then map to those we have already mentioned. The first is that of overseer (episkopos). The word has sometimes been translated as "bishop," but this translation is greatly misleading in our current context. A bishop today is a person of authority over many local church leaders, usually centered in a metropolitan area. The New Testament does not use the word episkopos in this way. A modern day bishop comes closer to an apostle in the early church than to an overseer or elder.
In the two words presbyteros (elder) and episkopos (overseer) we see the two principal ways of structuring church leadership today. "Presbyterian" churches tend to be governed by a local group of elders, while "episcopal" churches tend to have a hierarchy of leaders that govern many local congregations, perhaps even up to the level of "archbishops" who govern bishops. Over time, of course, these structures have multiplied in diversity. In actually, Congregational churches are even more "presbyterian" than the Presbyterians, for local Presbyterian churches are actually under an authority that goes above the level of the local church. "Congregational" churches, on the other hand, are self-governing on the local level, as are most Baptist churches.
The idea of an "episcopacy" that governs local churches within a hierarchy of regional leaders is the structure of the centuries. As early as AD110, the church father Ignatius was telling the church leaders of local congregations within a city to obey their bishop/overseer, who by that time governed an entire city of leaders. The shift back to a more congregational structure among some groups thus did not come into play until well into the Protestant Reformation.
Both groups can claim biblical precedents for their structures. The congregational/presbyterian format seems to reflect the structure of most local assemblies in the early church (although not the charismatic character of many in the earliest church). Meanwhile, the episcopal structure reflects the role that apostles played in the early church and the structure that God allowed to dominate the Church for 1500 years (although often not in the give and take of the earliest church).
When Paul addresses the leaders of the church at Philippi, he includes its "overseers" and "deacons." It does not seem likely that the church at Philippi, where Paul spent far less time than at Corinth, would be larger than the single church at Corinth. It would therefore seem likely that by "overseers," Paul is referring to the elders of the church at Philippi rather than to single individuals who governed individual churches.
At the same time, if the early church borrowed the model of elders from the synagogue, it is quite possible that it also borrowed the idea of a synagogue ruler. Here we are thinking of a person who took the leadership of a synagogue for a year or so and then perhaps passed it on to someone else at a later time. Just because early churches had councils of elders would not in any way preclude a single individual from serving as leader of the group. Indeed, it is hard to imagine such leaders not emerging, whether they had any official title or not. Once again, we do not find any particular structure among churches today eliminated. We only feel chastized if we have been dogmatic about our structure.
The role of deacon seems difficult to spell out. The seven appointees of Acts 6 are often considered the precedent for deacons as individuals who perform less spiritual functions within the church, like making sure widows get fed. But Acts 6 does not call these seven individuals deacons. The two we know the most about, Stephen and Philip, become preachers. Acts 21:8 calls Philip an "evangelist," and Acts 8 certainly shows him going around proclaiming the good news and facilitating the Holy Spirit's engagement with Judea and Samaria.
One possibility that immediately comes to mind is that deacons tended to be church leaders who were not old enough to be elders. For example, it is interesting that 1 Timothy, which tells Timothy not to despise his youth (4:12) also refers to Timothy as a "deacon," even if the word is not usually translated this way in 4:6. It is this word that is used of Phoebe in Romans 16:1. In any case, the frequent translation of "servant," in Phoebe's case joined to a word that has the sense of patron (Rom. 16:2), may suggest someone who supports the church in a somewhat concrete way.
In Paul's somewhat charismatic church world, he mentions many other roles a person might have in a church. 1 Corinthians 12:28 mentions apostles, prophets, teachers, and then blurs off into things like people who perform miracles or heal or administrate or speak in tongues. This is not an absolute list, as some to make up "spiritual gift tests" treat them. The first three have the most status, but then he mentions not offices in the church but a sampling of some gifts people in the churches of his day had.
So in Romans 12:6-8 he mentions prophecy, service, teaching, exhortation, contributing, leading, and being merciful. Ephesians 4:11 mentions apostles, prophets, evangelists, pastors, and teachers. What we have hear are important functions that need to take place in the church more than anything like an absolute list of how to package them. Those that expend great amounts of energy trying to line themselves up to these sorts of lists worry about lists Paul at least partially was creating on the spot in relation to his audiences and whose functions no doubt overlapped.
Evangelists were perhaps, like Philip, individuals who proclaimed the good news, but who had not seen the risen Jesus personallty. Travelling teachers would become a problem as the first century progressed. These were perhaps individuals who, like the Greek sophists, would set up shop in a place, relying on the patronage of some wealthy individual. A recurrent theme in books like 1 and 2 Timothy, Titus, 2 Peter, and Jude is false teaching, so it is no wonder Christianity soon developed a system of authority to control right teaching.
The role of prophets was apparently very significant in the early church. When Ephesians adds prophets to the foundation of the church (2:20), it is almost certainly referring to Christian prophets rather than the prophets of the Old Testament. 1 Corinthians 14 shows the important role they played in the worship at Corinth. Such individuals brought revelation to the assembly, which may often have urged the church on, and perhaps sometimes predicted things that were about to happen.
The vast majority of the functions of these "leadership roles" in the early church remain important for Christians today. However, there is no biblical mandate for us today to structure our leadership in the same way they did. Even in our descriptions of varying strands of the New Testament church, there is some variety. And even then, we do not find exact prescriptions for how to structure church leadership today.
 Indeed, some have used this instance as an argument against gambling, "casting lots." :-)
 Although some have argued that 2 Cor. 5:** implies that Paul had at least seen him while he was on earth. Paul's statement, however, does not necessarily imply that much.
 And while our personal inclination is to think Acts 14:23 is an accurate historical reflection of Paul's practice, to be circumspect we must acknowledge that Acts is not simply a documentary of what happened but is as much a position piece in relation to the church of its day, which we would see written probably in the 80s.