"But Jesus perceived their wickedness, and said, 'Why do you test Me, you hypocrites?'" (Matt. 22:18). He knows their intentions in spite of their kind words they spoke toward Him. They were hypocrites in that they didn't believe the things they spoke about Him, and they really weren't interested in what He had to say; their aim was to trap Him in His words!
Jesus then stated - "Show me the tax money." At that time, tribute for religion was paid in shekels (a Jewish coin), but tribute for Caesar was paid in denarii (a Roman coin). Jesus emphasizes the fact that the coin brought to Him was a Roman one by asking - "Whose image and inscription is this?" Jewish coins did not include images of men. Also, the fact that the Jews were using Roman coins was undeniable evidence that the Romans had dominion over their land. A denarius had an image of the head of the emperor stamped upon it and an inscription indicating who it was (in this case, Tiberias Caesar).
"Render therefore to Caesar the things that are Caesar's, and to God the things that are God's" (Matt. 22:21). The first part of the answer would satisfy the Herodians, and who could argue with the general principle that Caesar (or any human government) should receive the things which belong to it? The second part of the answer would satisfy the common people, and who could object to giving God that which is due Him? Essentially, Jesus answered their question regarding the lawfulness of Roman taxation with a "yes," but He made it very clear that paying such was not inconsistent with maintaining complete allegiance to God. God had long ago ceased to be the civil ruler of His people (cf. I Sam. 8:7), and there was nothing inherently wrong with a government taxing its subjects.
Human laws have no right to infringe upon the laws of God, but if they do, we must obey God and joyfully accept the consequences (cf. Acts 5:29,41). When there is no conflict between civil requirements and God's commands, Christians are to obey the civil authorities. This includes paying taxes (cf. Rom. 13:1,7). As a side note, it is interesting to consider the tax money as belonging to Caesar since it bore his image; likewise, humans themselves belong to God, for they are all made in His image (cf. Gen. 1:26,27).
"When they had heard these words, they marveled, and left Him and went their way" - They were amazed at His answer and found nothing therein to quibble about. He had escaped their dilemma masterfully, and they knew it. Likely they were also in awe of His insusceptibility to flattery, His immediate knowledge of their plot, and His loyalty to Caesar (since they believed that He was intending to establish a physical kingdom).