This morning I was reading Judges 19 and 20, the story where some Benjamite men of Gibeah committed a very wicked act, and all the other tribes of Israel gathered together and warred against them, killing them as punishment. Fortunately, we Christians aren't called to war against and kill other Christians who engage in wickedness. But today's verse lays out the steps we are to take when someone is living in unrepentant sin. Sadly, the instructions are often mishandled or outright ignored to the detriment of the church.
The first step after learning your brother has sinned is to address him in private, to avoid public embarrassment. Use scripture to make your case, and do it in love with the intention of him repenting. If this private confrontation doesn't work, only then do you include other people, and just one or two. Again, the purpose is repentance by the affirmation of two or three biblically minded individuals. Finally, if step two fails, you bring the issue before the entire church. Jesus wants His church to be a bride without blemish, and to do that we cannot allow sin to be swept under the rug where it can fester and grow. It will be embarrassing if someone's sin is made public, but by this point the offender has had multiple chances to repent, and they are now being given one final chance before being excluded from the assembly. It is a harsh, but necessary final step.
I once had a kid in my cabin at church summer camp, whose father had walked out on him, his brother, and his mother in order to enter into an adulterous relationship with a new woman. When my friend's mom went to their family's church with this, expecting them to respond biblically, they said that the man was still welcome, and that it wasn't their place to "tell people how to live". My friend's mom ended up taking them to a new church. What a sad situation! The Lord's church has a responsibility to call out and deal with this sort of wickedness among its members.
Some final notes: first, a mistake that is often made is taking others' sin public before doing step one, addressing them privately. This is gossip. Secondly, this method of confronting and excommunicating sins isn't for unbelievers (1 Cor 9:5-13). Finally, if someone gossips to you about another's sin, that gossip needs to be addressed as well.
Although it can be uncomfortable, I hope you and your church obey these steps outlined by Christ himself. When we do these out of order, or fail to do them at all, it dishonors God by failing to handle the sin in our midst.