What does Peter mean with the instruction to “honor the emperor”?

The word translated here as “emperor” is the common word for “king” (and ought to be translated as such; also in 1 Peter 2:13). When the emperor of Rome via his armies conquered new territory, he commonly did not dismantle the existing authority structures but instead used the structures to administer the defeated territory for him. So if the locals were used to a “king,” Rome commonly appointed a man to rule as king (Matthew 2:1; Acts 26:19). This man was responsible for collecting taxes for Rome, carrying out Rome’s dictates, developing policies relevant to the local province and enforcing them. The locals obviously saw more of this “king” than they did of the emperor in far-off Rome. In brief, he was the face of government and so the first recipient of whatever disdain people had for authority. Peter’s instruction now is that this man is to receive “honor,” be shown value, just as “everyone” was to be shown respect. And that would be true whether you were happy or unhappy about how (or even whether) he punished evildoers and praised those who do good (1 Peter 2:14).