"During the Vietnam war, I heard more than one preacher use the passage to defend the draft. Caesar (in this case the US government) has a right to draft people, and consequently Christians have a duty to "give unto Caesar that which was Caesar’s".
Another pastor, sometime later, used the same passage to justify tithing to the church (giving the church 1/10 of a one’s income)—based on the second half of the sentence.
Needless to say, neither of these reflects the meaning in the text itself.
The story there is that Jesus is approached by a group of Pharisees who have laid a neat trap for him. They ask him, publicly, whether Jews are permitted to pay taxes to the Roman government. If Jesus answers, “yes”, he will discredit himself with the Jewish community. On the other hand, if he answers “no” he will get himself in trouble with the Romans.
Jesus asks the Pharisees his own question. “Whose image is on this coin?” They answer, “Caesars.”
At this point, Jesus utters the famous sentence, confounding his enemies: “Give unto Caesar that which is Caesar’s, and give unto God that which is God’s”.” —Ken Watts [cont]