The Pharisees plot to destroy Jesus and devise a cunning plan to trap Him. They come with the Herodians and ask if it is lawful to pay tribute to Caesar or not. If Jesus says yes (which would please the Herodians) the Pharisees could stir up the people against Him since they hated the Roman oppressors. If He says no, the Herodians will run and report Him to the Roman authorities and they will execute Him for sedition. All this is done under a veil of respect and praise for the character of Jesus.
Jesus rebukes their hypocrisy and gives an answer they had not expected - that we are to give to God what is His and to the Roman authority what is rightfully theirs.
We apply this twofold message to us in this day and age. We may not like everything the government does but we do derive benefit from it and we are bound to submit to it as far as we are able, since it is instituted by God. We are also bound to give God what is rightfully His and all are called upon to examine themselves and see if they are faithfully serving the Lord with all that He has given them.