He is reputed as saying that we should render unto Caesar. Caesar seems to have an insatiable appetite for our renderings.
The Pharisees, or teachers of the law, were trying to trick Jesus into going against Rome so they could have him arrested. Judea was part of the Roman Empire, yet they were the only province that was allowed to keep their own religion instead of worshipping Caesar as a god. However, they were required to pay huge taxes (much of which would go into the tax collector’s pocket) to support the Roman government that they resented.
The Jews hated - absolutely hated - having to pay taxes to a polytheistic conquering state. Throughout the history of Israel, being free meant that God was with them.
When the Jewish leaders cornered him in front of the people, they were asking him a loaded question. They asked him if it was right for them to give money and tribute to pagan overlords, since the Roman mythology did not honor God. If he said that they should, they could rouse the Jewish people against him as a Roman sympathizer. If he said that they shouldn’t, they could turn him in as a zealot (one fo the Jewish “fanatics” that was trying to overthrow Rome with guerilla warfare).
Jesus asked for a Roman coin and put it back in the Pharisees’ court. He asked them whose picture was on the coin - it was that of Caesar. He told them to give to Caesar what was his - the coin with his likeness - and to give to God what was His - their hearts, their obedience to His Word. He was not only answering the question to the satisfaction of everyone involved, he was slapping the Pharisees down in public by spotlighting the fact that while they obeyed the letter of the Mosaic law, their hearts were turned toward themselves. How they looked to others was more important than how they looked to God.
That is the meaning of the saying - give to your worldly authorities what belongs to them, but give to God what belongs to Him.