In the Anglican Church, which follows the traditions of the Church catholic (from the Latin word meaning "universal"), there are three types of ministers or servants, bishops, priests, and deacons.

"Bishop" is an English word referring to what the New Testament (NT) in the original Greek language referred to as an episcopos, an "overseer."

As in the NT, the bishop is also an elder which in Greek is presbyter.

Further, the bishops and all other elders are all also deacons, from the Greek word diaconos meaning "servant."

So everyone who is ordained is a servant. Some are called to further service as elders. A few are called to even further service as overseers. Ordination is the formal process completed in a ceremony in which a person is charged with the responsibility of one of those roles. In the presence of the Lord Whom we rely on for His calling and equipping His servants, the gathered church then installs candidates in those roles.

So, where did the word "priest" come from? For some that evokes images of Old Testament priests who offered sacrifices. Importantly, the NT refers to every believer in Jesus as a priest, and all are called to offers sacrifices of praise and thanksgiving. But "priest" in reference to a clergy person in the Anglican tradition is a diminutive, a contraction, of the word transliterated from the Greek language, presbyter, an elder. As in the early Church, the bishop in each area, or diocese, is the focal point of unity in the Church in that diocese, one person serving in the stead of Christ. Then each of the other presbyters serves in the stead of the bishop in the various congregations in the diocese.

But why do catholic Christians call priests, "father"?

Jesus reproached the religious leaders of His day with some strong words about being called, "father," that sounds like we shouldn't use that name in the Church:

Mt 23:1-12  ESV Then Jesus said to the crowds and to his disciples,  2 "The scribes and the Pharisees sit on Moses' seat,  3 so practice and observe whatever they tell you - but not what they do. For they preach, but do not practice.  4 They tie up heavy burdens, hard to bear,1 and lay them on people's shoulders, but they themselves are not willing to move them with their finger.  5 They do all their deeds to be seen by others. For they make their phylacteries broad and their fringes long,  6 and they love the place of honor at feasts and the best seats in the synagogues  7 and greetings in the marketplaces and being called rabbi1 by others.  8 But you are not to be called rabbi, for you have one teacher, and you are all brothers.1  9 And call no man your father on earth, for you have one Father, who is in heaven.  10 Neither be called instructors, for you have one instructor, the Christ.  11 The greatest among you shall be your servant.  12 Whoever exalts himself will be humbled, and whoever humbles himself will be exalted. 

Jesus' words to His disciples on not calling anyone else "father" seems to make the catholic practice of addressing a priest as "father" completely unbiblical, a proud and Pharisaic thing to do. At least, a straightforward reading of the text might lead us to think that, what some would call a "literal" reading.

The problem with such a reading of this text is that it would also prevent Christ's followers from calling their earthly male parent, "father." Jesus is pretty explicit: " . . . call no man your father on earth . . ." The same thing applies to teachers, of course. We would have to cease calling all teachers by that name—Jesus doesn't restrict it to religious teaching—if one takes this statement at face value. But, human society and the Church within it still need parenting and teaching. Of course, if we stopped using those terms altogether it would be problematic, because then we wouldn't even be able to explain Who the heavenly Father is in relation to Jesus, or why Jesus alone is our Teacher, because those words would no longer apply to their earthly counterparts. No, Jesus wasn't teaching His disciples to stop using a name, a label, a way of referring to or addressing someone, whether "father" or "teacher." A literal reading of Jesus' words here just doesn't make sense. 

We run into further problems when we read the Apostle Paul's comments to the Corinthians and Thessalonians:

1 Cor 4:14-17  14 I do not write these things to make you ashamed, but to admonish you as my beloved children.  15 For though you have countless guides in Christ, you do not have many fathers. For I became your father in Christ Jesus through the gospel.  16 I urge you, then, be imitators of me.  17 That is why I sent you Timothy, my beloved and faithful child in the Lord, to remind you of my ways in Christ,2 as I teach them everywhere in every church. 

1 Th 2:11-12  11 For you know how, like a father with his children,  12 we exhorted each one of you and encouraged you and charged you to walk in a manner worthy of God, who calls you into his own kingdom and glory. 

One might argue that Paul didn't know Jesus had forbidden calling anyone on earth "father" but it is far more likely that he knew it and understood what Jesus meant, while it is our literal reading that doesn't accurately convey our Lord's meaning. 

The early Christians followed Paul's pattern. The Aramaic word abba ("father" or many would argue, "dad", e.g., Romans 8:15; Gal. 4:6) is the source of the English word "abbot" which the Celts first used for the head of their monastic communities. The French "Abbe" is similar, common for a parish priest. 

Oddly enough, the Christians that find it offensive to call a priest "father" routinely call their ordained church leader, "pastor." The latter term, however, is derived from the Latin word for "shepherd." Apparently, in their view, we have only one Father (as Jesus said) and so don't call their leader "father," but we have more than one shepherd. This is odd because Scripture is very clear on this:

Ps 23:1  The LORD is my shepherd . . .

But, it is even more specific. At one point during the ministry of the prophet Ezekiel, the Lord rebuked the shepherds of Israel, His flock:

Ez 34:2 "Son of man, prophesy against the shepherds of Israel; prophesy, and say to them, even to the shepherds, Thus says the Lord GOD: Ah, shepherds of Israel who have been feeding yourselves! Should not shepherds feed the sheep?  ...

So, the Lord then described how He would take over the role and shepherd (pastor) the people Himself, in statements like this:

Ez 34:11 "For thus says the Lord GOD: Behold, I, I myself will search for my sheep and will seek them out.  

If Psalm 23 wasn't clear enough, He makes it perfectly clear now:

Ez 34:23  "And I will set up over them one shepherd, my servant David, and he shall feed them: he shall feed them and be their shepherd."
Ez 37:24  "My servant David shall be king over them, and they shall all have one shepherd. They shall walk in my rules and be careful to obey my statutes."

One Shepherd, and yet we call our church leader, "Pastor" (shepherd)?

But, in fact, calling your church leader "Pastor" isn't wrong. For Christ is our "chief Shepherd":

1 Pet 5:1-4  ESVI exhort the elders among you, as a fellow elder and a witness of the sufferings of Christ, as well as a partaker in the glory that is going to be revealed:  2 shepherd the flock of God that is among you, exercising oversight, not under compulsion, but willingly, as God would have you; not for shameful gain, but eagerly;  3 not domineering over those in your charge, but being examples to the flock.  4 And when the chief Shepherd appears, you will receive the unfading crown of glory. 

Peter commands the leaders of the Church to "shepherd" the people of God, yet they have a chief Shepherd. With pastors (shepherds), it isn't that God doesn't use humans to do that work--He does--but they are only obedient and beneficial as God's shepherds if there is a clear understanding of who the real Shepherd is. 

Isn't this, however, exactly the case with fatherhood? We have a physical father, yet only one heavenly Father. We may have a spiritual father as well, perhaps several, yet understood as utterly insignificant in light of our heavenly Father because such "fathers" are only by analogy.  

To call our church leader "pastor" makes us sheep. Yes, we are likened to sheep in the Scriptures, and there are important lessons there for us. But, we never forget that we are people. And God's people live in families, not flocks. That is what the Church as a whole is, the family of God. Families have fathers. There is only one Father for the Church, the Father of our Lord Jesus Christ. But, for the church in this or that place, there is a leader, someone designated (ordained) to head up the family there. Isn't "father" actually a better term than "shepherd" for this head of the family? It is more intimate on a human level, closer to the truth of who we are-- people, now in God's family. 

So, why did Jesus say, "call no man your father on earth, for you have one Father, who is in heaven"? For the same reason He said, 

"If your right eye causes you to sin, tear it out and throw it away. For it is better that you lose one of your members than that your whole body be thrown into hell.  And if your right hand causes you to sin, cut it off and throw it away. For it is better that you lose one of your members than that your whole body go into hell."(Mt 5:29-30 ESV) 

He was making a point, a strong one, about the seriousness of sin. But, if we take it literally we might wind up killing ourselves to keep from sinning! Repentance is what Jesus was getting at, not self-mutilation. 

With giving our church leaders titles, whether "pastor", "teacher", or "father," Jesus' point is clear and driven home by the forceful way He expressed it: don't mistake anyone for the only real or true Shepherd, real Teacher, real Father that you have. 

So, yes, we call our priests, "Father Joe" or whatever their name is, but we don't make the mistake of thinking that means anything more than our desire to be a family together, to love one another as we have been commanded, so as to please our only true Father.