The English word "priest" is derived from the Greek word presbuteros, which is commonly rendered into Bible English as "elder" or "presbyter." The ministry of Catholic priests is that of the presbyters mentioned in the New Testament (Acts 15:6, 23). The Bible says little about the duties of presbyters, but it does reveal they functioned in a priestly capacity.
They were ordained by the laying on of hands (1 Tm 4:14, 5:22), they preached and taught the flock (1 Tm 5:17), and they administered sacraments (Jas 5:13-15). These are the essential functions of the priestly office, so wherever the various forms of presbuteros appear--except, of course, in instances which pertain to the Jewish elders (Mt 21:23, Acts 4:23)--the word may rightly be translated as "priest" instead of "elder" or "presbyter."
Episcopos arises from two words, epi (over) and skopeo (to see), and it means literally "an overseer": We translate it as "bishop." The King James Version renders the office of overseer, episkopen, as "bishopric" (Acts 1:20). The role of the episcopos is not clearly defined in the New Testament, but by the beginning of the second century it had obtained a fixed meaning. There is early evidence of this refinement in ecclesiastical nomenclature in the writings of Ignatius of Antioch (d. A.D. 107), who wrote at length of the authority of bishops as distinct from presbyters and deacons (Epistle to the Magnesians 6:1, 13:1-2; Epistle to the Trallians 2:1-3; Epistle to the Smyrnaeans 8:1-2).
The New Testament tendency to use episcopos and presbuteros interchangeably is similar to the contemporary Protestant use of the term "minister" to denote various offices, both ordained and unordained (senior minister, music minister, youth minister). Similarly, the term diakonos is rendered both as "deacon" and as "minister" in the Bible, yet in Protestant churches the office of deacon is clearly distinguished from and subordinate to the office of minister.
In Acts 20:17-38 the same men are called presbyteroi (v. 17) and episcopoi (v. 28). Presbuteroi is used in a technical sense to identify their office of ordained leadership. Episcopoi is used in a non-technical sense to describe the type of ministry they exercised. This is how the Revised Standard Version renders the verses: "And from Miletus he [Paul] . . . called for the elders [presbuteroi]of the church. And when they came to him, he said to them . . . 'Take heed to yourselves and all the flock, in which the Holy Spirit has made you guardians [episcopoi], to feed the church of the Lord.'"
In other passages it's clear that although men called presbuteroi ruled over individual congregations (parishes), the apostles ordained certain men, giving them authority over multiple congregations (dioceses), each with its own presbyters. These were endowed with the power to ordain additional presbyters as needed to shepherd the flock and carry on the work of the gospel. Titus and Timothy were two of those early episcopoi and clearly were above the office of presbuteros. They had the authority to select, ordain, and govern other presbyters, as is evidenced by Paul's instructions: "This is why I left you in Crete . . . that you might appoint elders in every town as I directed you" (Ti 1:5; cf. 1 Tm 5:17-22).