What Does the Bible Say About Church Discipline?

God wants us to discipline each other lovingly, for our good.

Matt 18:15-17 - If your brother sins, go and show him his fault in private; if he listens to you, you have won your brother. But if he does not listen to you, take one or two more with you, so that by the mouth of two or three witnesses every fact may be confirmed. If he refuses to listen to them, tell it to the church; and if he refuses to listen even to the church, let him be to you as a Gentile and a tax collector.

It is a sign of love to show someone a fault.

If my son is playing ball in a busy street, it shows no love for him whatsoever to allow him to keep playing in such danger. If I love him, I will warn him seriously. And if necessary, I will punish him appropriately.
In the same way, if I really love someone and see a spiritual fault in them, I'll notify them of their wrong. If I talk to others in deprecating ways about that fault, I reveal the falsity or shallowness of my supposed affection.
The result of telling someone their wrong is potential repentance and obedience. The result of talking behind someone's back about their wrong is a multiplication of sin - the continued sin of my friend, and now my own sin of not loving him enough to confront, or even showing disaffection for him by spreading gossip or slander.

It is a sign of concern for another's good to show them a fault in order to win them back.

My motive in taking my son aside should be to stop him from exposing himself to further danger, not to punish him just for punishment's sake or to show him who's boss.
The same principle holds true in the church. Our motive in taking someone aside to show him his fault should be to lead him to repentance, not to lord it over him. It should be for the brother's good - for the integrity of his verbal profession, for the reputation of his character, and for the reformation of his ways.