By Rabbi Carl M. Perkins:

Reproving Your Neighbor
Parashat Aharei Mot-Kedoshim (May 1, 2004)

You shall not hate your kinsman in your heart.
Reprove your neighbor, and incur no guilt because of him.
You shall not take vengeance
or bear a grudge against your countrymen.
Love your fellow as yourself: I am the LORD.
(Leviticus 19:17-18)

Why does it say “incur no guilt because of him”? You shall not shame him by reproving him in public. For it was taught that one who shames another person (“ha-malbin p'nei havero,” literally: one who whitens the face of another) is as if he sheds his blood.
(Rashi; Bava Metzia 58b)

Rabbi Haninah said: Jerusalem was destroyed only because people failed to reprove one another, as it is said: “Her leaders were like stags that found no pasture” (Lam.1:6) ­ meaning that just like stags, who stand with the head of one at the side of the other's tail, so too did the Jews of that generation hide their faces in the earth and did not reprove each other.
(Shabbat 119b)

One who reprimands his fellow ... must do so privately. And he should speak to him gently and with mild language, and let him know that he is only speaking to him for his own good, to bring him to the life of the world to come. If he accepts (the reprimand), all is well; if not, he should reprimand him a second and third time. And in fact a person is always obligated to reprimand him until the sinner drives him away and says to him, “I am not going to listen.”
(Maimonides, Mishnah Torah, Laws of Character Traits 6:7)

You shall not hate your kinsman in your heart … The verse here is saying the following: “Do not hate your brother in your heart when he does something to you against your will, but instead you are to reprove him, saying, 'Why did you do this to me?' and you will not bear sin because of him by covering up your hatred of him in your heart and not telling him, for when you will reprove him, he will justify himself before you [so that you will have no cause to hate him] or he will regret his action and admit his sin, and you will forgive him.”
(Nachmanides)

“Reprove your neighbor…” Literally, it says: Reprove, you shall surely reprove… Rabbi Ila said in the name of Rabbi Elazar the son of Rabbi Shimon: From this we derive that just as it is a mitzvah to speak when our words will be heeded, so it is a mitzvah not to speak when our words will
not be heeded. Rabbi Abba said: in such a case it is an obligation not to speak, as it is said: Do not reprove a scoffer, for he will hate you; reprove a wise man, and he will love you.”
(Proverbs 9:8)

When you want to reprimand someone, do not do so by calling them a scoffer! If you do, he will hate you -- Instead, reprove a wise person -- by saying: you are so wise, a person of your quality should not do thus-and-such... Then, the person will love you -- and listen to your words and accept your guidance.
(Sh’nei Luhot Ha-Brit, Rabbi Isaiah Horowitz, 16th-17th c., Prague and Tzefat)