From the Greek, it means to change mutually, thoroughly, fully - to make things different, and in the Hebrew - to cover over, forgive, wash away, atone.

In scripture, it's most important function is of our being reconciled to God. It is something that God desires for all creation, because he loves us, he does not like to see us at odds, hurting each other and ourselves. Jesus himself was our conciliator, and has now given us the job...

Reconciliation with God

Romans 5:8-11
But God demonstrates His own love toward us, in that while we were still sinners, Christ died for us. Much more then, having now been justified by His blood, we shall be saved from wrath through Him. For if when we were enemies we were reconciled to God through the death of His Son, much more, having been reconciled, we shall be saved by His life. And not only that, but we also rejoice in God through our Lord Jesus Christ, through whom we have now received the reconciliation.

Daniel 9:24
"Seventy weeks are determined
For your people and for your holy city,
To finish the transgression,
To make an end of sins,
To make reconciliation for iniquity,
To bring in everlasting righteousness,
To seal up vision and prophecy,
And to anoint the Most Holy.

Acts 10:34-43
Then Peter opened his mouth and said: "In truth I perceive that God shows no partiality. But in every nation whoever fears Him and works righteousness is accepted by Him. The word which God sent to the children of Israel, preaching peace through Jesus Christ--He is Lord of all-- that word you know, which was proclaimed throughout all Judea, and began from Galilee after the baptism which John preached: how God anointed Jesus of Nazareth with the Holy Spirit and with power, who went about doing good and healing all who were oppressed by the devil, for God was with Him. And we are witnesses of all things which He did both in the land of the Jews and in Jerusalem, whom they killed by hanging on a tree. Him God raised up on the third day, and showed Him openly, not to all the people, but to witnesses chosen before by God, even to us who ate and drank with Him after He arose from the dead. And He commanded us to preach to the people, and to testify that it is He who was ordained by God to be Judge of the living and the dead. To Him all the prophets witness that, through His name, whoever believes in Him will receive remission of sins."

2 Cor. 5:18-21
Now all things are of God, who has reconciled us to Himself through Jesus Christ, and has given us the ministry of reconciliation, that is, that God was in Christ reconciling the world to Himself, not imputing their trespasses to them, and has committed to us the word of reconciliation. Now then, we are ambassadors for Christ, as though God were pleading through us: we implore you on Christ's behalf, be reconciled to God. For He made Him who knew no sin to be sin for us, that we might become the righteousness of God in Him.

Ephes. 2:14-16
For He Himself is our peace, who has made both one, and has broken down the middle wall of separation, having abolished in His flesh the enmity, that is, the law of commandments contained in the rule books, so as to create in Himself one new man from the two, thus making peace, and that He might reconcile them both to God in one body through the cross, in this way putting to death the enmity.

Col. 1:20-22
and by Him to reconcile all things to Himself, by Him, whether things on earth or things in heaven, having made peace through the blood of His cross. And you, who once were alienated and enemies in your mind by wicked works, yet now He has reconciled in the body of His flesh through death, to present you holy, and blameless, and above reproach in His sight--

Ephes. 2:17-22
And He came and preached peace to you who were afar off and to those who were near. For through Him we both have access by one Spirit to the Father.
Now, therefore, you are no longer strangers and foreigners, but fellow citizens with the saints and members of the household of God, having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief corner stone, in whom the whole building, being joined together, grows into a holy temple in the Lord, in whom you also are being built together for a dwelling place of God in the Spirit.

The priest's job is to be a concilator for his people, and we are called to be kings and priests as Christians....Rev. 1:6; Rev. 5:10; 1 Peter 2:5; 1 Peter 2:9. The word used here (kaphar) is normally translated "atonement" and appears many times to do with God's desire and methods to reconcile people to himself...

Ezekiel 45:20
And so you shall do the seventh day of the month for every one that errs, and for him that is simple: so you shall reconcile (atone for) the house.

Exodus 30:10
And Aaron shall make an atonement upon the horns of it once in a year with the blood of the sin offering of atonements: once in the year shall he make atonement upon it throughout your generations: it is most holy unto the LORD.

Exodus 30:15-16
The rich shall not give more, and the poor shall not give less than half a shekel, when they give an offering unto the LORD, to make an atonement for your souls. And thou shalt take the atonement money of the children of Israel, and shalt appoint it for the service of the tabernacle of the congregation; that it may be a memorial unto the children of Israel before the LORD, to make an atonement for your souls.

Leviticus 4:26
And he shall burn all his fat upon the altar, as the fat of the sacrifice of peace offerings: and the priest shall make an atonement for him as concerning his sin, and it shall be forgiven him.

Hebrews 2:17 Wherefore in all things it behoved him to be made like unto his brethren, that he might be a merciful and faithful high priest in things pertaining to God, to make reconciliation for the sins of the people.

Reconciliation brings the unity of the church, which brings us to Godly perfection, completeness and maturity, able to do the work of God. Col 3:14 describes the bond of peace, is the ligament or uniting principle that joins the body together...

Not the unity of uniformity, but of one goal under Christ. General Patton said that if all are thinking alike, no-one is thinking at all. This war we are in demands all of our individual and collective God-given faculties working to their utmost... It is a great general who can cause this to happen.

Col. 3:14-15
But above all these things put on love, which is the bond of perfection. And let the peace of God rule in your hearts, to which also you were called in one body; and be thankful.

1 Cor. 10:16-17
The cup of blessing which we bless, is it not the communion of the blood of Christ? The bread which we break, is it not the communion of the body of Christ? For we being many are one bread, and one body: for we are all partakers of that one bread.

1 Cor. 12:4-10
Now there are diverse gifts, but the same Spirit. And there are differences of administrations, but the same Lord. And there are diverse operations, but it is the same God which works all in all. But the manifestation of the Spirit is given to every man to profit from. For to one is given by the Spirit the word of wisdom; to another the word of knowledge by the same Spirit; To another faith by the same Spirit; to another the gifts of healing by the same Spirit; To another the working of miracles; to another prophecy; to another discerning of spirits; to another diverse kinds of tongues; to another the interpretation of tongues:
1 Cor. 12:12-28
For as the body is one, and has many members, and all the members of that one body, being many, are one body: so also is Christ. For by one Spirit are we all baptized into one body, whether we be Jews or Gentiles, whether we be bond or free; and have been all made to drink into one Spirit.
For the body is not one member, but many. If the foot shall say, Because I am not the hand, I am not of the body; is it therefore not of the body? And if the ear shall say, Because I am not the eye, I am not of the body; is it therefore not of the body? If the whole body were an eye, where were the hearing? If the whole were hearing, where were the smelling? But now hath God set the members every one of them in the body, as it has pleased him. And if they were all one member, where were the body? But now are they many members, yet but one body. And the eye cannot say to the hand, I have no need of you: nor again the head to the feet, I have no need of you. No, much more those members of the body, which seem to be more feeble, are necessary:
And those members of the body, which we think to be less honourable, upon these we bestow more abundant honour; and our uncomely parts have more abundant comeliness. For our comely parts have no need: but God has tempered the body together, having given more abundant honour to that part which lacked: That there should be no schism in the body; but that the members should have the same care one for another. And whether one member suffer, all the members suffer with it; or one member be honoured, all the members rejoice with it. Now you are the body of Christ, and members in particular.
And God has set some in the church, first apostles, secondarily prophets, thirdly teachers, after that miracles, then gifts of healings, helps, governments, varieties of languages.

Ephes. 4:1-6
I therefore, the prisoner of the Lord, beseech you that ye walk worthy of the vocation wherewith ye are called, With all lowliness and meekness, with longsuffering, forbearing one another in love; Endeavouring to keep the unity of the Spirit in the bond of peace. There is one body, and one Spirit, even as ye are called in one hope of your calling; One Lord, one faith, one baptism, One God and Father of all, who is above all, and through all, and in you all.

John 17:23
I in them, and thou in me, that they may be made perfect in one; and that the world may know that thou hast sent me, and hast loved them, as thou hast loved me.

Psalm 133:1-3
A Song of Ascents. Of David.
Behold, how good and how pleasant it is
For brethren to dwell together in unity!
It is like the precious oil upon the head,
Running down on the beard,
The beard of Aaron,
Running down on the edge of his garments.
It is like the dew of Hermon,
Descending upon the mountains of Zion;
For there the LORD commanded the blessing--
Life forevermore.

And reconciliation between people in general, as individuals, as groups, as nations.

Matthew 5:23-26
Therefore if you bring your gift to the altar, and there remember that your brother has something against you, leave your gift there before the altar, and go your way. First be reconciled to your brother, and then come and offer your gift. Agree with your adversary quickly, while you are on the way with him, lest your adversary deliver you to the judge, the judge hand you over to the officer, and you be thrown into prison. Assuredly, I say to you, you will by no means get out of there till you have paid the last penny.

Col. 3:10-15
And have put on the new man, which is renewed in knowledge after the image of him that created him: Where there is neither Greek nor Jew, circumcision nor uncircumcision, Barbarian, Scythian, bond nor free: but Christ is all, and in all. Put on therefore, as the elect of God, holy and beloved, bowels of mercies, kindness, humbleness of mind, meekness, longsuffering; Forbearing one another, and forgiving one another, if any man have a quarrel against any: even as Christ forgave you, you also do. And above all these things put on charity, which is the bond of perfectness. And let the peace of God rule in your hearts, to the which also you are called in one body; and be thankful.

Because enmity is a work of Satan

Genesis 3:15
And I will put enmity
Between you and the woman,
And between your seed and her Seed;
He shall bruise your head,
And you shall bruise His heel."

Rev. 6:4
Another horse, fiery red, went out. And it was granted to the one who sat on it to take peace from the earth, and that people should kill one another; and there was given to him a great sword.

God desires families to be united, fathers to their children. So much pain has resulted in war - which has both physically taken the male influence from the family, and encouraged women to be the major influence at home on the children, and also replaced men's position as sole provider. Children are twisted and neglected. Most troubled children are from the homes of solo parents. This is not an indictment on those who are in such a hard position - it is just to say that children really need both influences in their upbringing. It is a needed role in the church for men to offer some time to such children.

Malachi 4:5-6
Behold, I will send you Elijah the prophet before the coming of the great and dreadful day of the LORD: [6] And he shall turn the heart of the fathers to the children, and the heart of the children to their fathers, lest I come and smite the earth with a curse.

Luke 1:17
And he shall go before him in the spirit and power of Elias, to turn the hearts of the fathers to the children, and the disobedient to the wisdom of the just; to make ready a people prepared for the Lord.