I will briefly give you the Scriptural teaching on bishops, elders, and deacons, and then I will tell you what happened historically. I will verify the historical teaching with readily available quotes (see sidebar) so that you don't have to take my word for it.
Paul and Peter—the only two New Testament authors who use the terms—use bishops (or overseers, Gr. episkopoi) and elders (gr. presbuteroi) interchangeably. They also tell these elder-overseers that they are the ones who will shepherd (or pastor) the church of God. (The word "overseers," used in most Bibles is the same word we translate "bishop.")
If you hear teaching about this subject, that's about the extent of what you'll hear. The New Testament says overseers and elders are the same men, and they are the shepherds of the Church.
While that's true, it's not so simple.
We have letters dated not long after the apostle John died from a man that John appointed as an overseer. His name is Ignatius, and he was bishop in Antioch, the apostle Paul's home church. Of course, the apostle Paul had been dead for about 50 years when Ignatius wrote his letters. John outlived Paul by 30 or 40 years.
Tradition—tradition that most historians consider reliable—has it that the apostle John appointed Ignatius as overseer.
It is clear from Ignatius' letters that John's church leaders did not have the same roles as Paul and Peter's church leaders. Antioch, Ignatius' home church, and the churches to whom Ignatius wrote (except Rome, which both Paul and Peter helped establish) each had one overseer and multiple elders. Their elders were not all bishops. Only one church leader was the overseer.
By the mid-2nd century or so, it seems clear that every well-organized church functioned this way, and almost all Christian writers of the 2nd century and later seem to have forgotten that there was ever a time when churches had multiple overseers.
When I address whether there was a pope in the early Church, I will give you clear, strong evidence that Rome didn't have a singular bishop until at least A.D. 120. That evidence is so clear and obvious that no one would doubt it if the Roman Catholic Church were not around to fight for their doctrine of the primacy of the bishop of Rome.
For now, what I have just written is not controversial. The Scriptures from Peter and Paul are there for you to see, and the history of John's churches is simply what's accepted. It's awful hard to argue with Ignatius' letters and other evidence.
The term "priest" came to be used as the western churches began to use Latin in their writings during the 3rd century.
The Greek word for elder, presbuteros, was simply brought into Latin as presbyter. This should really carry the connotation of elder, not priest, but mid-3rd century Latin writers, like Cyprian (a great and godly man despite the fact that I consider this a doctrinal fault), also referred to elders with the Latin word sacerdos, which is one of the several Latin words for a priest, a person who offers sacrifices.
The church had grown by the 3rd century, and thus the leaders tended to carry more responsibility than a century earlier, when a family atmosphere with more share responsibilities was more likely. Both Justin (A.D. 155) and Tertullian (c. A.D. 200) refer to the leader of a Sunday meeting as "the president." Since these are the only two places that title is used, it is very likely that it means "whoever happens to be presiding at the meeting." The bishop or elder was not necessarily present.
Remember, church buildings would not have been possible in Justin or Tertullian's times. They would have been very temporary structures, surviving only till the next persecution, when they would have been taken or destroyed. Therefore, what is likely is that many meetings were done in houses that could not contain the whole church, necessitating multiple meetings with leaders that were not elders.
By Cyprian's time (A.D. 250), however, the battle against heresies had led to a stronger emphasis on meetings led by ordained (and thus trusted) persons. Since the meetings involved offering sacrifices of praise, prayer, and devotion of themselves to God, as well as the offering up of the bread and wine in communion, the elders were seen as offering sacrifices from the church to God.
In this way, they gained the term priest.
As the church grew, the power of the overseer grew as well. Ignatius led a congregation that was much smaller than the ones led by subsequent bishops in Antioch. (Like Theophilus, who left us his epistle, To Autolycus, in A.D. 168.)
As churches grew in size, the brotherhood in smaller towns often did not have their own bishop. Instead, the bishop of the nearest large city would serve as overseer of the whole area. Overseers with this kind of territory were called metropolitans.
By the time of the Council of Nicea in A.D. 325, the bishop of Alexandria ruled over all of Egypt and the Roman overseer over an undisclosed area. Here's how the Canon from the Council of Nicea reads:
These three metropolitans became known as patriarchs. Later, the bishop of Constantinople was declared a patriarch as well. Centuries later, a patriarch was also added in Russia.
The four patriarchs of Alexandria, Antioch, Constantinople, and Russia are in fellowship with one another to this day. Political separation after the fall of the western Roman empire in the 5th century kept the Roman patriarch separate from the others for centuries. When political boundaries were overcome, the patriarchs were unable to heal their separation, and in the "Great Schism" of 1054 the separation was finalized, resulting in the official existence of the Roman Catholic Church (as separate from the eastern "Orthodox" churches).
The other four patriarchs are the heads of the Eastern Orthodox Church (Constantinople), the Greek Orthodox Church (Antioch), the Coptic Orthodox Church (Alexandria), and the Russian Orthodox Church. They maintain a shared leadership over Catholicism in their respective areas to this day.
Obviously, things are a little more complicated than that. With travel the way it is, you can attend a Russian Orthodox Church right here in America, in the supposed territory of the bishop of Rome. However, what I have just described is the basic history of the development of bishops, metropolitans, patriarchs, and the pope.
Rome would dispute this, of course, but the Orthodox churches would not. The history is clear. The Roman Catholic Church is simply imagining a history that includes a pope before Nicea.