Polity refers to the government of the church. Although there is no reason to think we are limited to the forms of polity used in the New Testament era, it is instructive to see how they ran their churches as a beginning point.
How were New Testament churches led?
There were two offices evident in the New Testament church: elders and deacons.
Elders
Elders vs. Bishops: The office of elder is synonymous with the office of bishop. (See Acts 20:17,28 and Titus 1:5,7 where the two terms are used interchangeably). "Elder" comes from the word presbeuteros which means an older man, and therefore describes a person who is relatively mature spiritually. "Bishop" comes from the word episkopos which literally means an overseer. Therefore, this term describes what the person does (i.e. oversees the local church).
The New Testament church appears to have consistently established a plurality of eldership in each local church.
- Acts 14:23 - ". . . appointed elders (plural) in every church (singular). . . "
- Titus 1:5 - ". . . appoint elders (plural) in every city (singular). . . "
- 1 Pet. 5:2 - ". . . shepherd the flock (singular) of God among you (plural). . . "
The qualifications for elders
These generally fall into two categories:
He must be functionally effective in spiritual leadership
Just as Jesus said sheep would know the voice of their shepherd, (see John 10:4), it seems likely that those considered for eldership in the early church had already demonstrated the ability to lead. This is probably why Paul waited for a while after starting the churches in Ephesus and Crete before he had Timothy (ch.3) and Titus (ch. 1) appoint elders.
It took time for the true leaders to naturally emerge.
The ability to lead others in spiritual matters is also implied by the fact that the elder must be "skilled at teaching" (1 Tim. 3:2), or, "able to exhort in sound doctrine and refute those who contradict" (Titus 1:9). A teacher is not skillful unless his students learn. Learning includes how to do God's will, not just how to know it. (James 1:22-25) Finally, elders and deacons had to "hold fast the mystery of the faith with a clear conscience" (1 Tim. 3:9) which would not be possible if they were guilty of sins of omission (see also "above reproach" 1 Tim. 3:2).
All of these references imply that elders had to be practitioners of the Word, not just theoreticians.
He must be morally upright and consistent.
The emphasis in the Bible is on character even over and above function. You can study a list of the qualifications for elders along with suggested definitions for each.
Deacons
Likewise, Deacons were required to have functional and character requirements before they could serve. You can study a list of qualifications for deacons along with suggested definitions.