The Presbyterian model does not tend to see a distinction between these classes, though particular Presbyterian churches may differ in the details. This is founded in (what the Reformers believed to be) the practice of the early Church, according to evidence from the Bible and from the Church Fathers.
- Deacons are still around. But they are not trainee priests, or assistant priests, or anything like that; their role is pastoral or administrative or both.
- Everybody is a priest, but only some people are ministers.
- The unique functions of bishops are distributed among various assemblies.
The priesthood is typically held to be common to all believers; the relationship between us and God is not interrupted by any intermediate layer. However, there is still a function of ministry to which certain people are called:
The duties of ministers are various; yet for the most part they are restricted to two, in which all the rest are comprehended: to the teaching of the Gospel of Christ, and to the proper administration of the sacraments. 1, Ch. 18
Ministers are meant to be associated with a particular local congregation. No minister has any more authority than any other, and in particular, "bishop" is taken to be a term that puts emphasis on the administrative or governing role of the minister, rather than denoting a distinct class.
Now the one and an equal power or function is given to all ministers in the Church. 1, Ch. 18
In giving the name of bishops, presbyters, and pastors, indiscriminately to those who govern churches, I have done it on the authority of Scripture, which uses the words as synonymous. 2, IV.3.8
All, therefore, to whom the office of teaching was committed, [the early Church] called presbyters, and in each city these presbyters selected one of their number to whom they gave the special title of bishop, lest, as usually happens, from equality dissension should arise. The bishop, however, was not so superior in honour and dignity as to have dominion over his colleagues ... And the ancients themselves confess that this practice was introduced by human arrangement, according to the exigency of the times. 2, IV.4.2
In the Catholic system, bishops do many different jobs. They are in charge of their diocese, they exercise discipline over the local priests, they ordain new priests and assign them to parishes, and so on. In the Presbyterian system, these tasks are the responsibility of an assembly/presbytery/session/synod instead (whose members are ministers and elders - representatives of congregations). The same jobs are being done, but it's no longer a single man doing all of them. For example, the local congregation will "call" a minister to the parish; he is not sent there by the bishop. The authority of these assemblies does not come from a notion of apostolic succession; it's simply a matter of what is considered an appropriate way to make decisions.
That leaves deacons. There are some differences between various churches here. Deacons may be chosen as administrators (looking after church finances), or as having a special pastoral office directed towards the poor, or both.
There will be two classes of deacons, the one serving the Church by administering the affairs of the poor; the other, by taking care of the poor themselves. 2, IV.3.9
[Deacons in the early Church] received the daily offerings of the faithful, and the annual revenues of the Church, that they might apply them to their true uses; in other words, partly in maintaining ministers, and partly in supporting the poor. 2, IV.4.5
1 The Second Helvetic Confession. Heinrich Bullinger, 1562.
2 Institutes of the Christian Religion. John Calvin. 1559. Translated by Henry Beveridge, 1599.