When considering who to nominate to be an elder of our church, it is important to consider the biblical requirements and qualifications for an elder.

An elder is one who serves the body of Christ by shepherding the body through the oversight of the church and the care of souls and by providing godly wisdom and a godly example  that can be followed. (1 Pet 5:2-3)
Note: Biblically, "elder", "overseer", and "pastor" are terms all used to describe a single leadership office in the church.

While it may seem a basic requirement, it is a critical one: a man must desire, be eager, and be willing to be an elder. (1 Tim 3:1; 1 Pet 5:2; Heb 13:17)

Elders are to live lives of godliness that can be followed and to shepherd the body of Christ as Christ would have them. Thus, there are certain character qualities that an elder must have. What is extraordinary about these character qualities is how ordinary they are. As D. A. Carson says, the character qualities expected of an elder are "remarkable for being unremarkable." They are the character qualities that we should expect of any good Christian.

Here's the list:

1. Above reproach (1 Tim 3:2; Titus 1:6,7)
2. Husband of one wife (1 Tim 3:2; Titus 1:6)
3. Sober-minded (1 Tim 3:2)
4. Self-controlled (1 Tim 3:2; Titus 1:8)
5. Respectable (1 Tim 3:2)
6. Hospitable (1 Tim 3:2; Titus 1:8)
7. Not a drunkard (1 Tim 3:3; Titus 1:7)
8. Gentle, not violent (1 Tim 3:3; Titus 1:7; 2 Tim 2:25)
9. Not quarrelsome (1 Tim 3:3; 2 Tim 2:24)
10. Not a lover of money/greedy for gain (1 Tim 3:3; Titus 1:7; 1 Pet 5:2)
11. Manage his own household well (1 Tim 3:4-5)
12. Keeping his children submissive in a dignified way (1 Tim 3:4)
13. Children are believers and not open to the charge of debauchery or insubordination (Titus 1:6)
14. Not a recent convert (1 Tim 3:6)
15. Well thought of by outsiders (1 Tim 3:7)
16. Humble, not arrogant (Titus 1:7; 1 Pet 5:5; Matt 10:25-28; Mark 10:42-45)
17. Not quick-tempered (Titus 1:7)
18. A lover of good (Titus 1:8)
19. Upright (Titus 1:8)
20. Holy (Titus 1:8)
21. Disciplined (Titus 1:8)
22. Be an example to the flock (1 Pet 5:3; Heb 13:7)
23. Kind (2 Tim 2:24)
24. Patient (2 Tim 2:25)

Able to Teach
The one ability that is explicitly stated is the ability to teach (1 Tim 3:2; 2 Tim 2:24) specifically according to biblical doctrine (Titus 1:9). It is important to note here that "able to teach" does not mean "able to preach."Preaching is a particularl form of teaching and proclamation that some men are gifted at (to varying degress) while others are not. "Able to teach" is about the ability to expalin biblical doctrine (particularly the gospel) and give wise godly wisdom from that doctrine (both positive instruction and protective refutation of bad doctrine). This does not mean that an elder will know the depths of every doctrine. Rather, it means the elder is convinced and committed to fundamental doctrines that are core to the gospel and following Christ. "Teaching" may occur from the pulpit, in the classroom, in a small group, or even one-on-one. It may be formal (think sermon or bible study) or informal (think counseling and one-on-one conversations).

A helpful article to understand this more is "What 'Able to Teach' Means and Doesn’t Mean" by Sam Emadi.

Able to Lead and Manage Well 
But another implicit one is the ability to lead and manage well. (1 Tim 3:4-5; Acts 20:28)

An elder is expected to not just lead but to lead like Christ. This means that the tactics and approaches of the secular world (e.g. in business or even other non-profits) are only appropriate as much as they align with the way Christ leads (1 Pet 5:2-3; Matt 20:25-28; Mark 10:42-45) . This is why the character traits of an elder are those of an "ordinary" Christian: he must be like Christ to lead like Christ. This is to be expected as the flock that the elder shepherds is not his flock by God's flock (1 Pet 5:2; John 21:15-17; Acts 20:28).

It's important to note two overarching realities about an elder's wife (assuming he is married and not single):

1. There is NOT an office of an elder's wife. (There is only an office of an elder.)
2. An elder's wife occupies a unique position in the body due to her closeness to an elder.

It is safe to infer from Scripture that an elder's wife is to be a Christian. This can be inferred from the fact that an elder is to manage his own household well and he is not to be unequally yoked in his marriage. This does NOT mean that an elder's wife must or is entitled to run certain ministries or serve the church. Rather, it means that she should be a follower of Christ who  seeks to be like Christ.

Being an elder's wife can be taxing as she will have to practice a great deal of discretion and forgiveness in order to support her husband and not unnecessarily harm his ministry as an elder or harm the church. It is important she is supportive of her husband's ministry, the other elders, and of the ministry of the church. A great deal of damange can be done through criticism and a critical spirit.

This three-part series from biblicaleldership.com on Elder's Wives is helpful:

1. Questions from Elders’ Wives, Part 1
2. Questions from Elders’ Wives, Part 2
3. Qualifications for Elders’ Wives