What does Luke 6:27 mean?

This is one of the most difficult and counter-cultural commands in the Bible. Understanding the various terms in context is helpful. An "enemy" is someone who does not like you and wants to harm you. "Love" is from agape and means to seek someone else's welfare, even if to do so requires a sacrifice. Likewise, "good" means to act for the advantage of another. "Good" is from the same root word as "well" in Luke 6:26. Sinful people will speak for the benefit of other sinful people who provide them with worldly advantage. Jesus-followers will act for the benefit of others, even those who mean harm (Luke 6:22).

It makes no human sense to work toward the best for someone who actively wants to cause hardship. This is only possible, in an emotionally healthy way, if our focus is not on our enemy but on our God and His promises. It is Jesus who died for the world while we were His enemies (Romans 5:10). Because of Jesus' sacrifice, God promises if we do love our enemies He will give us great reward, not least of which is being called His children (Luke 6:35).

Context is important, however. This passage comes after an explanation that God will bless those who suffer because they follow Christ. He will judge those who are praised by the world because they say what the world wants to hear (Luke 6:22, 26). The enemies, here, are those who hate those who love Christ. Likewise, the context of this command is in seeking the other person's good—not enabling, endorsing, or cooperating in their abuse or evil.