What does Jesus mean when he says to turn the other cheek?

do not repay evil for evil. Overcome evil with good.

He means that when you wrestle with a pig you both get dirty, and the pig likes it.

FORGIVE THE PERSON WHO wronged you.

As you would expect. If someone strikes you on the left check, offer that person the right cheek. That is if the abuse or persecution is in Jesus' name. If someone is attacking you, for no apparent reason other than to kill or harm, you have the right to defend yourself.

Turning the other cheek does not mean that a Christian would not defend himself against violent assailants.

Jesus was not saying that we should never defend ourselves but, rather, that we should never strike offensively, that we should not allow ourselves to be provoked to take revenge.

While it is wise to retreat whenever possible in order to avoid a fight, it is proper to take steps to protect ourselves and to seek the help of the police if we are a victim of a crime.

Matthew chapter 5, verse 39. There Jesus says: Do not resist him that is wicked; but whoever slaps you on your right cheek, turn the other also to him.

Did he mean that if someone hits you with his fist on one side of your face, you should let him hit you on the other side?

No, Jesus did not mean that at all. A slap is not like a hit with the fist. It is more like a push or a shove. A person may slap us to pick a fight. He wants us to get angry.

And if we get angry and we push or shove back, what happens?

We will probably get into a fight. Jesus is not here referring to a blow intended to do injury but to an insulting slap with the back of the hand.

But Jesus did not want his followers to get into fights. So he said that if someone slaps us, we should not slap him back.

We should not become angry and get into a fight. If we do, we show that we are no better than the one who started the fight.

If trouble starts, what do you think is the best thing to do? It is to walk away.

The other person may push or shove you a few more times. But that will probably be the end of it.

When you walk away, it does not mean that you are weak. It means that you are strong for what is right.

But what if you let yourself get into a fight and you win? What could happen then? The one you beat up could come back with some friends.

They might even hurt you with a big stick or a knife. So, can you see now why Jesus did not want us to get into fights?

Be a peace maker .. love the other person. Luke 6:27-29 ESV
“But I say to you who hear, Love your enemies, do good to those who hate you, bless those who curse you, pray for those who abuse you. To one who strikes you on the cheek, offer the other also, and from one who takes away your cloak do not withhold your tunic either.
Romans 12:19 ESV /
Beloved, never avenge yourselves, but leave it to the wrath of God, for it is written, “Vengeance is mine, I will repay, says the Lord.”

There was no jesus.

When you strike an inferior, you backhand them, which for right handed people, means the other's right cheek. If you then present the left cheek, they have to hit you with the palm of their hand - as you would do to an equal. So, while you do not fight back, you are demanding that they other treats you equally. Passive resistance! Basically, the message is to not escalate the violence, but, do not cower either.

No revenge.
No anger.
Compassion and kindness.

This was Jesus telling His followers how to respond to personal insults. They were to quietly take it, and even stay around for more insults. Which is why I'll not be leaving this site any time soon.

The verse in question (Matthew 5:39) requires verse 38 to be understood first. Jesus was correcting a wrong perspective many Jews had about the Mosaic Law that said "an eye for an eye and a tooth for a tooth" was required when meting out justice. This had been turned into exacting revenge but that's not what the Law meant. It was a way of saying no more, and no less than the exact equivalent was to be rendered in justice. But people wrongly assume that it speaks of taking vengeance. It does not. Therefore, Jesus said that if an evil person is aggressive towards them, they must not resist him but turn the other cheek, offer their tunic, go the extra mile with them.

The Christian attitude shows strength under control. One controls their temper and maintains a loving attitude even if this appears as weakness to ungodly, aggressive people. That's what Jesus meant, and He exemplified that superbly.

Be humble

This was a key part of the Christian religion for their slaves that the Romans invented. Christianity was designed as a pro-Roman form of Judaism that would not only reduce Jewish support for the troublesome Zealots, but the religion would also help to keep the slaves more docile. The Roman's idea worked very well.
https://www.amazon.com/Caesars-Messiah-Conspiracy-Flavian-Signature/dp/1461096405